I think I'm okay with flatting preflop to set mine.

As for the flop, I think betting was not the right move for exactly the reason you mentioned. The odds that one of them holds an Ace is high. If your reasoning for flatting preflop was to set mine, why would you make a 3/4 size bet on the flop?

Even if their check seems to show weakness, it doesn't mean that you should c-bet because most of their range is beating you (35 out of 47 possible holdings beats you) Only the hands that beat you will call.

AA, AKo, AQo, AKs, AQs, AJs, KK beats you
QQ chops
JJ, TT you beat


As for the river, I think everyone else covered it quite well.