The US could do away with the Electoral College and it would still be a republic.  Who know's when you flunked 8th grade?

This jumps out at me from your Hamilton link:

"A small number of persons, selected by their fellow-citizens from the general mass, will be most likely to possess the information and discernment requisite to such complicated investigations."

So Hamilton thought the people needed elites to pick their president for them. Back in Hamilton's day, only white men who owned property were allowed to vote. But even that was too much popular democracy for him. The choice of president was to be filtered further:

"And as the electors, chosen in each State, are to assemble and vote in the State in which they are chosen, this detached and divided situation will expose them much less to heats and ferments, which might be communicated from them to the people, than if they were all to be convened at one time, in one place."

Oy ve!