The Pinkney plan, short form:
1) assign seats to a Federal legislature lower house, the Assembly, proportionally to each state.
2) Annually State legislatures shall select their apportioned number of Delegates tp the national Assembly.
3) The Delegates shall choose replacements for the outgoing Senators. Senators belong to one of 4 classes, and the nation is divided into 4 large districts of equal population, each one associated with one of the classes. Each Senator has a 4 year term, and the classes are staggered so one is up for reelection every year. Thus every new Assembly has 1/4 of the Senate to fill, all chosen from the people of one of the districts.
4) the Assembly and Senate together vote, presumably one vote for each, to elect a President for one year.
My comments:
This system divorces the Federal government from any direct control by the people at large. The closest they come to being involved is by electing their state legislatures, which take it from there. This obviously means that any political enthusiasm among the masses, or tendency to expand politics to a universal adult suffrage, will be focused on state politics primarily, and in that sense States Rights reign supreme. In the other "draft" Pinckney submitted in the 1810s of his plan however, the Delegates were to be elected by the people directly, and this makes a huge difference--completely bypassing state power and making the Federal government a true union. But it is doubtful Pinckney himself would have advocated for this in 1786!
If we assume the version with state legislatures appointing the Delegates prevails, it is an open question what policy the individual legislatures use to assign their annual share of Delegates. The simplest would be winner take all, with the victorious party within each legislature sending only its own delegates, leaving all others who may have won some legislative seats out in the cold as far as that state's voice is concerned. I believe they would tend to gravitate toward a more proportional split of the Delegate seats, the lion's share going to the dominant party but some remnant being allocated to the second running party, and maybe to all parties making a large enough showing. Why? Because over the years it will become obvious that pretty much the whole weight of government falls on the Assembly of Delegates, and drastic overturns of its entire membership every few years are not truly desirable; someone in the Assembly should have long tenure and learn to handle the environment there masterfully. Each party will begrudge the other even existing, judging by OTL experience, and at first favor winner take all and a total housecleaning, at least of each state's delegation when the "good guys" win control of their legislature at long last, but over time it will become clear that if the "good guys" want to be effective in the Assembly they need to preserve a core of career Delegates, and that they need to agree to grant the other side a proportional or at least token share of Delegates they control in order to conserve their own professional, career Delegates, and so that they are obliged by quid pro quo, ultimately I suppose enshrined in each state's laws or constitution, to let the other party have their minority share in turn. Of course two dominant parties might easily agree to freeze out any third parties, who would have to gain control of the legislature completely to get any Delegates at all, unless some states have been liberal enough to make their delegation truly proportional all along, or eventually. A third party gaining control of a legislature in a state that has no law requiring fair representation for the second party might well vindictively take all the seats to itself, but it would probably be more in their interest to make their state law mandate proportionality.
Assuming at least some states do mandate either two party splits or true proportionality, this means a certain number of Delegates are reliable for each party in each of these states, even in years they do badly in the state legislature, and so since the seats are not tied to particular districts, just a total count, each party can designate a list of individuals in order wanted in the Assembly, so that the top one and probably typically the entire top quarter to third remain in office year to year, despite annual elections. To be sure with the legislatures individually in charge of how they assign their Delegates, some states might continue or revert back to winner take all, and others might try experiments like dividing their state into districts and letting the people in each one vote winner take all, allowing them to thus elect their Delegates directly. Still I think that a class of professional Delegates, who owe their tenure in the Assembly to loyally serving the will of a party in their home state, will arise and these will tend to dominate and guide the Assembly.
The Assembly, if run by professional Delegates, is the center of all Federal power. The President in essence is the creature of the Assembly majority. Senators in theory have independence for 4 years, and a say in the election of the President too, but any Senators wishing to keep the position past 4 years must position themselves to please the majority of the Assembly in their election year. A 4 year term does not seem like much to me to set the Senators very far apart or "above" the Delegates. In fact it might be a clever maneuver on the part of opposition in the Assembly to kick a rival upstairs, to take advantage of a Senatorial class election to place a rival Delegate leader of another party into the Senate, thus removing them from the Assembly for the next 4 years!
I can imagine that perhaps this plan might be accepted more or less as is. At first to someone of Hamilton's mentality it must seem outrageous, but if he comes to realize the likelihood that a professional class of long term Delegates with seniority will arise, and that by mandating states to appoint their Delegates proportionally to their win in the legislature, or alternatively in true proportion to the public vote cast directly for Delegates should a state enact that option, without creating Delegate districts, then he might come around, realizing it is a Trojan Horse for the sort of central state he envisioned.
If accepted, I suppose George Washington would keep getting reelected for years after 1789, and might have to forcefully remove himself from consideration by 1796, maybe earlier. Note that creating the "great districts" for the Senate, when there are 4 of them, could be approached by finding the north-south balanced division of states by population, and then dividing each subgroup again evenly, either north/south again or preferably if possible, east/west. The boundaries of these quadrants would shift--Pinckney did not mention a Census but either someone will or it will belatedly be addressed by Assembly/Senate Congressional legislation--I don't believe the President has any veto power. It is obviously necessary to survey the state populations periodically to keep their assigned number of Delegates proportional to changing population ratios; the Censuses, whether ad hoc or regular, are the time to shift Senate district boundaries between the states. Because half the districts will be southern initially, half the Senate will come from slave states. Pinckney proposed both the 3/5 compromise and extension of the Articles of Confederation full faith and credence clause to all felonies, so laws like the fugitive slave acts will probably follow. I'd have to look up the language again but it is proposed to allow either 2/3 of Assembly and Senate each to agree to admit a new state, or 2/3 of the state delegations in block to agree, or 2/3 of state legislatures, not sure which, but anyway since unified resistance of each sector can block the other from getting new states admitted, there will also be various Compromise acts balancing admissions between the sectors. Events like the Texan secession from Mexico seem most likely to happen, along with things like the Louisiana purchase. The latter might be stymied by the manifest inability of the weak President to accept Napoleon's author without checking with the Assembly first, but a very bold President might confidently assume the Assembly would back them for such tremendous gains at such a low price, so we could look at versions where the Purchase fails and those where it succeeds anyway. Eventually civil war will loom and there is nothing in Pinckney's proposal to provide for secession any more (or less!) than the OTL constitution. However the only way for a party like the Republicans to dominate, and that would be for them to win a majority in a majority of state legislatures. With proportionality they might win overwhelming majorities in northern states and maybe with the aid of small minorities in some Southern states tip the balance, but they would still have 3/4 the Senate elected by older parties and once or twice in the next several years be forced to choose Senators from the slave states, so there might be a long delay past 1860 before anti-slavery Congressional majorities choosing an anti-slavery President come about--the USA might wind up neck and neck with Brazii for last big nation state to abolish slavery!
Assuming the USA can get past this somehow and wind up with slavery abolished, I don't see any fundamental impediment to the nation being run much along OTL lines, broadly speaking. With the President elected by Congress (the official name of the combined Assembly and Senate, something like "USA in Congress Assembled" would no doubt be truncated in colloquial use to just "Congress" I suppose) every year, as with the Assembly being run by the same politicos reelected every year by their parties in their various states with fluctuating levels of backbencher neophytes to bulk out their proportional strength and recruit replacements for fired, retired or kicked upstairs to Senate stalwarts, that when a given party dominates the Assembly and Senate combined they keep reelecting the same guy with no limits unless he gets burned out and refuses, observes the traditional Washington example of 6-9 years being enough, gets dropped for disappointing performance or the party loses its majority and is replaced by the other party's man, so in reality Presidents would serve fairly long terms, probably averaging 5-7 years with extremes running to maybe 15-20 and a fair number of 1-2 year wonders on the other end of the spectrum. But since the President is utterly beholden to Congress Assembled for renewal of mandate every year, he is the creature of his party just as the Senators are, and again it is the Assembly that is the center of mass of power. It may well be that the public loses sight of this and judges parties by their Presidents, but it will be painfully clear that the only way to influence who the President is is to support the party that offers him as their proposed candidate in one's own state-or run around other states trying to influence their legislative outcomes of course. States that allow the public to directly vote in Assembly members district by district (Assembly districts will not be a standard national thing but a state option, and perhaps one ruled out by amendment or law or the original Constitution as it emerges from the Convention) will tend to lose influence in the Assembly, unless districts return the same guys year after year, which they certainly might. If adopting the policy of requiring the state legislatures to seat Assembly members in proportion to the outcomes of legislative elections, or allow the public to vote them in with proportional representation directly as an alternative, become the norms, then a half-assed form of PR applies to the Assembly, and hence perhaps the Senate indirectly, across the nation.
Anyway if Americans keep this system but are shrewd about it, they will recognize that what they have is a kind of Parliamentary system, albeit with a weirdly ephemeral and short-time ranged House of Lords analog that will not be excluded from legislation because mutual veto of the two houses is written into the Constitution and because the "upper" house is really just a time-delayed echo of the lower one, no matter how much pretense there is it has a higher dignity. Power resides mainly in the party-chosen leader of the party that dominates the Assembly, and everyone else, even the President, defers to this leader--or not, at their peril, unless they recognize that guy is losing his mandate. Overall then a general pattern similar to OTL in policy may prevail--the Federal government will gain power regardless of Pinckney's intents; the public will change policy by shifting support to a given party and away from failing leaders of last year, but since support for a party is tied up with lots of policy on various levels and with tradition for many voters, there will be a lot of inertia; unless PR is adopted as the norm in electing the state legislatures there will probably be strong two party dominance which might be reinforced by customary or legislated practices (such as only dividing the Assembly delegation proportionally between the top two parties for instance). There are wider possibilities but probably to make them normal some rules have to change.
So, after a Civil War period we could expect the South to resent freeing the slaves and Jim Crow to follow; for the dominant antislavery party to shift over to become the party of corporate cartels since these lay behind the rapid expansion and maturation of American business in the late 19th and early 20th century, for socialist and other populist radical parties to get lost in the thicket of layered practices favoring two parties but to influence the platforms of the dominant ones into a balance of power between conservatives oriented toward the successful big business supporters versus crusading Progressive reformers, for Imperialism and "a Navy second to none" to appear as a bipartisan win-win the conservative wing and not a few reformers align around and for the USA to consequentially become more adventurous in Latin America, east Asia and ultimately the world, and therefore become entangled in big foreign wars overseas. During wartime conventional wisdom would say not to mess with the President, as long as he is delivering a defensibly decent progress toward victory anyway, and the norm of "politics stops at the shoreline" holding both parties to parallel and cooperative foreign policy planks, with the opposition party held to support the in party's foreign policy actions with only limited and careful criticism. A welfare state could also emerge I would think, and stuff like NASA.
And much of this is down to the fact that given a tendency for parties to form and to keep returning a core of stalwart partisans to Assembly year after year, and to back the same guy for President, possibly having a generation where two leaders alternate in executive power much like Gladstone vs Disraeli in the later Victorian years, many of Alexander Hamilton's goals would be met despite the apparently radical opposition in concept to Hamilton's openly centralized and authoritative vision.
OK, that is my take on adoption of Pinckney's scheme. Let's see if I can be more succinct about Hamilton!
{A note from an earlier draft I included by mistake, on proportional appropriation of Delegates to states}
...the state legislatures, not direct votes of the public, would select Delegates to an Assembly, apportioned proportionally by means of providing one for each "N" thousand in the state (with the 3/5 compromise included to count each slave by 60 percent for purposes of apportioning Delegates). The "N" represents something in the revisionist document Pinckey himself submitted years after the Constitution went into force, a placeholder for the convention to agree upon had they adopted his plan--he clearly did not mean to recommend one Delegate for every thousand of population however counted! Had that approach to proportionality been adopted, it would be achieved by tremendous numbers of delegates even if N were reasonably large, for N would have to be less than however few thousands of population the smallest state would have. If the plan were generally adopted, I suspect this would first morph into something closer to OTL apportionment policy--every state gets at least one, some scheme to apportion the remaining seats to the states proportionally would be adopted and revised as necessary by Assembly and Senate legislation, as would the total number of seats in both Assembly and Senate. Under P's plan there is no linkage between the number of states and number of Senators, but the Delegates must be at least close to proportional to population.
OTL, Jefferson devised a method that is equivalent mathematically to iterative formulas, Hamilton devised a rather simpler method that tends to award slightly more seats to the smallest entities (which I favor for proportional seating of candidates, for one thing because calculation is relatively easy and easy to explain and justify, but don't like so much for apportionment of seats to states) and Daniel Webster later devised another method basically similar to Jefferson's but with a slightly closer to proportional outcome. All of these have been used by Congress at one time or other but have since been replaced by a third iterative method (unlike the others, useless for proportional elections) that is said to achieve closest proportionality, that is to say the least deviation possible given the rule that every state gets one. Any of these would be acceptable for apportionment I think though the latest method would be adopted eventually I suppose. The method of just saying "one for every X thousand" will run into trouble fast and either it will be replaced before the Constitution is adopted, perhaps as OTL vague language demanding proportionality as the goal but leaving implementation up to Congress, or at a later date an Amendment of some kind (Pinckney proposed mechanisms for calling a new Convention, and I am unclear on whether he also had a second track for Amendments via Congressional proposal and state legislature ratification as OTL too) would revise the method.