Skallagrim
Banned
To make this work, I think you need to tweak the pre-Revolutionary history a little. I think you need more colonial aristocrats, who are on side with the rebellion. And/or you need more involvement from either the French Canadians or some kinds of native confederacies, who believe themselves to have some kind of special relationship with the monarchy and who are thought will cause problems if they don't get a king.
A less drastic alteration might be to 'just' escalate Shays' Rebellion, so that a much more urgent need for strong and stable leadership is widely perceived. In light of what @Galba Otho Vitelius wrote earlier...
One additional point about Hamilton is that New York had withdrawn its delegation from the constitutional convention. Hamilton stayed on, so New York could be said to "participate", but really had no standing. In other words, he probably shouldn't have been there at all and his views likely to be disregarded for reasons other than being radical. His importance is exaggerated because of his writings to convince the New York convention to (narrowly) ratify the Constitution anyway.
...a particularly handy 'tweak' would be to have the rebellion spread to New York, so that after its defeat New York is suddenly all the way on board for a strong national executive, and Alexander Hamilton will have a lot more clout at the convention. That would probably make for a good POD here.
What Hamilton seems to be proposing is a President for Life who could be removed by Congress fairly easily (or resign) short of that. The critical part is the circumstances under which removal "for good behavior"is possible, and to a lesser extent the process of election.
For example, in modern day South Africa, the President -who is not elected for life but for renewable terms like elsewhere- is both elected by Parliament and is removable by Parliament, by a simple majority vote for any reason. One President has been removed in this manner, and it involved no suspicion of criminality or other type of misconduct. As political scientists have pointed out, this makes the President of South Africa more of a Prime Minister who happens to moonlight as the head of state. The point is that if you make it easy to remove the President, the ultimate effect is not very monarchical!
One big blind spot of the framers of the 1787 Constitution its hat they seem to have thought that the federal government could and would operate without political parties. That has to be kept in mind whenever you see super-majority requirements written into the Constitution. That, and the fact that both Houses of Congress had much fewer members than we are used to in the 20th century (the House settled at 435 members in 1910 and the Senate at 96, to increase later to 100). The 1790 Senate, after Rhode Island finally joined the union, had 26 members, for example, with a reasonable expectation of a half a dozen more being added shortly. With a 32 member Senate, only 21 Senators gets you a two thirds majority. So under the provisions of the Constitution, impeachment and removal of the President required only House majority and 21 Senators, certainly nothing close to the 67 at present, and the situation where the President's party would block impeachment for partisan reasons, something that happened IOTL at both presidential impeachment trials, was just not anticipated.
Astute observations! One thing we may be sure of is that Hamilton didn't intent for an executive that would be easy to remove. If the idea got to the stage of being refined, one may be assured he'd plead for distinct supermajorities in both houses being required to impeach/dethrone the chief executive.
To be clear, I absolutely agree with you that - barring a clear pre-convention POD - this kind of scheme isn't likely at all to be adopted. Hamilton's plan was just way to far removed from what most people had in mind, and George Washington was too much of a Cincinnatus to go for it, plus the fact that Hamilton had no formal say, as you pointed out. Yet it remains interesting, and with a good POD, we could butterfly all three objections away.
I envision a worse Shays' Rebellion, escalating to new York, forcing Washington to personally lead an army to stop the rebels (much like the Whiskey Rebellion in OTL). Ideally, his personal prestige leads the rebels to peacefully surrender to him, allowing him to grant blanket amnesty and see the peace restored. It doesn't get much more kingly than that, does it? After that, with the need for a strong executive starkly clear, the whole mood for the convention is different. New York is all for a strong executive suddenly. Several leading politicians explicitly beg Washington to fill that role. The recent troubles play a big role in convincing Washington that his country needs him. And then at the convention, Hamilton proposes his plan in a well-prepared way (as I outlined earlier), and then...
...well, then we're all set.