I have to wonder if these people who parrot the "republic not a democracy" talking point actually know the context.
Way way back in the late 18th century, "democracy" meant "direct democracy" (think ancient Athens) and "republic" meant "representative democracy" (think all democratic countries today--Switzerland's probably the closest to a direct democracy, but there are important checks on the power of the voters as a legislative body). Athenian democracy was essentially mob rule, and Madison et al were desperate to avoid that (see, for instance, Federalist No. 55). Hence they set up a republic (ie a representative democracy) specifically to hold the power of the mob in check. (Hence why the President is elected indirectly and the Senate was initially also elected indirectly.)
So technically the people who say "republic not a democracy" are right, if you're 225 years behind in your definitions.