Every definition of "republican form of governance" involves some level of democracy. It's not "direct democracy", but it is "representative democracy".
In the "good old days" there were some offices where the people voted for state representatives who in turn selected federal representatives, but even in these cases there was some role for citizen voters to play in who represented them.
Anyone who says that the US is a republic and not a democracy doesn't really understand what the word "republic" means. If you want to completely eliminate democracy from our governance system then you are a true Rightist who wants us to go back to a monarchy or similar.
Of course the reality is that we are neither a direct democracy nor a republican representative democracy. For some time we have been an oligarchy ruling over state fiefdoms. So niggling over what type of democracy we have is beside the point.
BS