Being a republic, and a democracy aren’t exclusive of each other.
That is correct. In fact, a Republic is a kind of democracy, a representative democracy. Yes, a Republic is not a “pure democracy” or “direct democracy” where the people directly implement policy. Rather, a Republic is an indirect democracy where the people’s authority is exercised through elected representatives, but that is still a kind of democracy.
The notion that a Republic is not a democracy is one of the more popular myths out there.