Strictly speaking, the early us was more of an Aristocratic Republic rather than a democratic republic.
> The House of Representatives shall be composed of members chosen every second year by the people of the several states, and the electors in each state shall have the qualifications requisite for electors of the most numerous branch of the state legislature.
Article 1, Section 2.
In the Early US, only white male land owners could vote (the upper class, aka, the aristocracy). Inasmuch, it was not what most people would consider a democracy. The lower class had no representation whatsoever.
So, when you say that a "A republic is a kind of democracy" it's actually not true, at least not for what most people would consider democracy.
It was a bit wider than that, the US was notable at the time for a highly unusual degree of land ownership. After all, there was a lot of land to steal from the native americans.
Regardless, the US was a democracy before women could vote, and it was a democracy before black people could vote, and it was a democracy before non-landowners could vote. It was a democracy in which many inhabitants didn't truly have full citizenship, but to this day there are millions and millions of people who live in the US (including many citizens!) who cannot vote, and it's still a democracy.
