The following opinion piece has some interesting background on things like what the Founding Fathers meant with democracy, as well as the historical origin of "This is a Republic, not a Democracy".
https://www.nytimes.com/2019/08/27/opinion/aoc-crenshaw-republicans-democracy.html
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In more modern terms, the founders feared “direct democracy” and accounted for its dangers with a system of “representative democracy.” Yes, this “republic” had counter-majoritarian aspects, like equal representation of states in the Senate, the presidential veto and the Supreme Court. But it was not designed for minority rule.
Virtually everything was geared toward producing representative majorities that could govern on behalf of the country — to diminish “faction” in favor of consensus. And in the case of the Electoral College, the point wasn’t to stymie majorities but to provide a way to find a competent and popular chief executive in a large nation of parochial states.
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These origins are important. If there’s substance behind “We’re a republic, not a democracy,” it’s not as a description of American government. There’s really no difference, in the present, between a “republic” and a “democracy”: Both connote systems of representation in which sovereignty and authority derive from the public at large.
The point of the slogan isn’t to describe who we are, but to claim and co-opt the founding for right-wing politics — to naturalize political inequality and make it the proper order of things. What lies behind that quip, in other words, is an impulse against democratic representation. It is part and parcel of the drive to make American government a closed domain for a select, privileged few.