I agree with you on "republic", in fact I have said the same many times.
Democracy however doesn't necessarily have to be of the Athenian model. All that is required for democracy is that the people are the source of decision making.
This is what Plato tells us: The core of democracy, metaphysically speaking, is "rule by freedom". The idea of human equality comes from the idea of universal human freedom. You can only avoid doing what your better tells you to do, if you are instead their equal.
If I were going to use my own words, and be precise, I would call the United States, in it's beginning, a mixed system. It was designed to mix plutocracy with democracy.
Remember that in the beginning, the United States had property rules as to who could vote.
Nowadays, the government straddles democracy, and tyranny. All democracy eventually devolves into tyranny, so this is not surprising.
What we SHOULD be doing, instead of playing politics.