“Republic” (any system in which top-level government offices aren't personal property of the officeholder) and “Democracy” (any system in which government serves and is accountable to the general population—usually through voting on candidates and/or specific policies—rather than vice versa) are not mutually exclusive.
It’s pretty common for modern Western governments to be both of (Democratic, often also Federal) Republic and (Representative, sometimes with minor areas of Direct) Democracy.
To put a finer point on it: actual experts in the study of government routinely refer to the US as a democracy. It’s absolutely not a sign of better familiarity with the topic to “correct” that usage—it’s a sign of low side of middling familiarity, specifically.
Debate whatever you want, we do not have a direct democracy, which is what most people hear when they hear "Democracy". It's Representative Democracy. Supposedly.
That isn’t what most people think of when they hear “democracy” in the context of describing countries like the US as a democracy. Lay-usage and expert usage are in accord here. It means more-or-less liberal and with voting that significantly affects how the state runs and/or who runs it. That’s all, and that usage doesn’t confuse anyone. If we didn’t use “democracy” for that we’d just have to come up with something else, because it’s a very useful term to have. But everyone just uses “democracy” and that works great.
It's a federated representative democracy with some issues, some of the components of which have some level of direct democracy.
It is also a republic. It could be a republic even if it was a pure direct democracy somehow without "issues" - in fact I struggle to see how a direct democracy without issues could be something other than a republic.
It’s pretty common for modern Western governments to be both of (Democratic, often also Federal) Republic and (Representative, sometimes with minor areas of Direct) Democracy.