How democratic was the Roman Republic? I understand it was a slave society, so the slaves were obviously not enfranchised. What about free Romans, who weren’t enslaved? Did the average person’s life change much when the Senate lost power and Augustus became emperor? And what, if anything, does it say about the US that people on both sides of the aisle are fond of citing the fall of the Roman Republic as a parable on the fragility of democracy?

Depends what period of the Republic you’re talking about. There were periods where the constitution changed to be more democratic, after the secessions of the plebs for example; there were periods where the constitution changed to be less democratic, such as after Sulla made himself dictator and rewrote the constitution to disempower the plebians. 

As to how the average person’s life changed, it’s a bit tricky to say, in part because Augustus maintained the forms of the Republic, so that the Senate retained a good deal of influence for a long time. (Arguably even after the fall of Rome itself). 

As to people in the U.S, that speaks to the enduring classical trend in American constitutional thinking since before the American Revolution.