I actually tend to think a) it's quite true that America was founded as a "republic" and not a "democracy" and b) that was a mistake.

There's no doubt that Madison preferred the "aristocratic" republics like Venice. Yes, many people during the founding also used the term "democracy" to refer to a representative democracy, and Madison's view was not the only one at the time.

I also completely agree: Read the Federalist Papers! Discover that the main worry about "factionalism" was to defend creditors and the boss from debtors and the poor, and that Madison never explains how to defeat minoritarian factions

Learn that the Founders thought institutions had to be designed to counteract ambition and selfishness but that somehow senators would have Ciceronian public virtue! Witness all the incoherencies and forced arguments to square their elitism with America's democratic spirit