By 1820 the Federalists, as a presidential party, had simply disappeared. A small state cast three electoral votes for John Adams so that Washington's unanimous elections would remain unique. Monroe's election was otherwise unanimous.
The reason for the Republican (no, not today's Republicans) hegemony was "simple"- the Republicans had adopted all of the Federalist positions.
Certainly, in today's world, a Republican will never be legitimately elected President unless they have recognized the challenge of global warming. Sadly, this does not rule out another Republican President who does not recognize the problem.