the stage where it happens

Hamilton depicts the presidential election of 1800 as evidently a three-man race among Jefferson, Burr, and Adams, with Adams a sure loser.

That's not what happened at all, though it does set up the story's historically authentic dilemma in which Hamilton is forced to choose between Jefferson and Burr, and supports Jefferson.

Let me see if I can explain briefly what did happen.

In 1796, the parties had been disciplined in their vote for President (same as in 1800, Adams v. Jefferson), but their votes for VP had been scattered. But, as the show notes, officially the electors' two votes were undifferentiated for office. As a result, though Adams had been elected President, no running mate had more votes than Jefferson the opposition presidential candidate did, so he had become VP.

In 1800, both parties were determined to do it better. Each agreed on a ticket and stuck with it. The Federalists, Adams for re-election with C.C. Pinckney for VP; the Republicans, Jefferson for President and Burr for VP. As the show notes, Hamilton loathed Adams, and he went to some trouble to try to arrange a situation where Pinckney would get more electoral votes than Adams and become President instead if the Federalists won. But none of the other Federalists much liked the idea of deposing the incumbent, and Hamilton's idea went pffft.

The Republicans were aware of the problem of an equal number of electoral votes for the ticket, but Jefferson and Madison were reluctant to offend Burr by withholding any votes from him. They just hoped it would happen in some state that voted Republican but that they didn't have any control over. But it didn't. Everybody stuck to the ticket.

Incidentally, only a few states chose electors by popular vote. In most states, they were chosen by the legislature, making legislative elections the popular vote proxy for presidential ones. The New York legislative election earlier that year, then, was when Hamilton and Burr as party managers faced each other down at the street level, even running into each other at polling places. Burr was victorious, and that's what made Republican victory in the presidential race likely, though it was by no means assured. Despite the show's claim, the Federalists were still very much a live option.

Once Jefferson and Burr had equal electoral votes, the election by constitutional rule moved to the House of Representatives to break the tie. The Republicans stuck to Jefferson. Though Hamilton announced for Jefferson, the Federalists in the House ignored him and voted for Burr. But, despite the show's claim, at no time did Burr openly campaign for President against Jefferson. He merely said he would abide by the result, neither campaigning nor withdrawing. He might have been doing much behind the scenes, but we don't know.

Neither candidate had a majority for several ballots, because they voted by state delegation and some were tied. Eventually the Federalists realized they weren't going to get Burr into office against firm Republican support for Jefferson, and gave up.

And that's how Jefferson became President. It wasn't Hamilton who did it. He had very little influence in this election, even with his own party. But he did earn Burr's enmity, so there's that.