A representative democracy is a subset of all democracies, which are a subset of all republics. What the Federalist Papers talks about is the danger of a Direct Democracy. Seriously, read it sometimes. Don't take the word of others for what's in them.
So yes, we have a democracy. We also have a republic. Sometimes, I suspect that this right-wing talking point was created because some Republicans were tired of Democrats sounding like they're the only ones for a democracy, since their name is so close.
But even a Republic's financial solvency can be threatened by a majority of politicians out-promising each other over how many gifts they will give people in order to get elected.
Representatives handing out cash in return for elections is pretty much exactly what is happening now. No different than people directly voting for more money for them.
The fiscally conservative side of the voting block is very concerned that a numb