However, there's a third possibility: what if separate elections were held for the presidency and vice-presidency? What would US politics be like if you could, for instance, vote for a Republican presidential candidate and a Democratic vice-presidential candidate? (Or whatever - the POD for this would be early enough that the political parties might be different).
A great question.
A large part of the reason for the system as proposed in 1787 was the Framers lack of experience with national politics of any sort. Up until that time the only kind of politics had been conducted in colonial assemblies and then state legislatures; the discussion of the Continental Congress and the Congress of the Confederation were much more diplomatic in nature, between representatives of various governments. Combine that with the presumption of the time that political factions were bad and the Founders didn't do a good job of designing an electoral system.
However, they also spent a very, very short amount of time considering the executive branch (about 2 weeks versus 2 months spent on the Legislature). The most insightful (in terms of creating the American system of checks and balances) device was to ensure a separately elected executive branch, rather than one chosen by Congress. The original electoral college, which presumed there would be no national political culture, but a varied array of state factions, was not without merit, if one conceives as the choice not which of two (or more) platforms/factions you would rather see govern and more as "name your two most trusted individuals." IMO this is more than just "Washingtonian optimism," but the nature and development of American political parties is tangential.
OTL the Senate of the First Congress chose to deprive John Adams of any real authority as Vice-President because Adams was "obnoxious and disliked" (to quote
1776). He advocated giving the POTUS formal titles so as to better face-off against foreign monarchs. And he tried to speechify in the Senate to the acrimony of many.
The merits of a separately elected VP are fairly apparent today, IMO, but are essentially comprise a boost to legislative-executive checks and balances. Many states (Georgia and Texas) have separately elected Lieutenant Governors. In both cases the Lt. Gov derives his power from having real power as President of the Senate to set a legislative agenda. Indeed, in many ways this is more power than the President has in domestic policy. Such a system results in a much less powerful executive branch, but also more interesting political groupings.
Grey Wolf has proposed in his "Joubert 1799 and Carnot 1801 TL" a system in which Alexander Hamilton as VP for Charles Pinckney in 1804 is the progenitor of the an ATL Committee system. OTL this was the system devised by Henry Clay that empowered the office of Speaker of the House of Representatives. It never applied to the VP because the Senate had already ruled the powers of the VP as President of the Senate to be constrained to breaking ties and breaking ties alone. The key to ensuring a strong VP is a committee system that empowers the VP to arrange the composition of political power and preference within the Senate, rallying coalitions and shaping the details of legislation. Indeed, the VP would have more power in doing so for being independent of any coalition of support within the Senate. Say both the VP and the Speaker had power to appoint Committee members; say the composition of both houses was more or less equal (by %), and both parties had 2 or more major coalitions. The Speaker would appoint committees and the like based on the coalition of support that elected him, uniformly from his own party. The VP, however, could appoint whom he pleased. Say he was a moderate Republican: he might appoint some moderate Dems as Committee Chairs. He is far more free in chosing his committees than the Speaker is. Accordingly, even though the composition of both bodies is roughly equal, the legislation proposed and approved by each will have a different slant, requiring all sorts of compromise.
An independently elected VP would essentially create useful friction between the President, the VP, and the Speaker, complementing the friction between the Executive as a whole, the Senate, and the House of Reps. However, the insight and structures necessary to such a system requires 1) more elaboration about the nature of executive power and 2) more elaboration about the powers of the President of the Senate. The fact that so very many of the powers the VP could exercise--is he member of the Cabinet?--are not formal parts of the OTL 1787 Constitution (or the Constitution as amendend as practiced today).
An interesting question is that an indepedently elected VP might not present such a strong incentive towards a two-party system as that posed by the OTL electoral college. Two parties could form a coalition by agreeing to support one's candidate for Pres and the other's for VP. More likely, it will emphasize coalitions within Parties rather than just Parties. This year, for example, I think a significant number of Americans would try to vote for John McCain and Joe Biden, if afforded the opportunity. This could be very interesting, since a more multi-partied USA will also see more Presidential elections decided by the Congress, a system that could be mutually reinforcing.
There will be an additional complication in today's environment: an odd facet of the way modern primaries work is that they almost replicate the orginal structure of the electoral colelge. Every one campaigns for President and the runners-up hope they might get to be VP. Indeed, one of the oddest (IMO) aspects of the primary system is that the VP is almost chosen by fiat by the nominee, with little input from the Party. This was not the case in the heyday of Party conventions, when candidates were almost always decided by floor vote. TTL's Conventions, both modern and prior, will be complicated by an indepednetly elected office. The ability to form a balanced ticket will be undermined, since the electorate can always chose a new one. You might have situations like OTL's Election of 1824 in which John Calhoun was the VP of both John Quincy Adams and Andrew Jackson. The potential for chaos is much more pronounced.