Hey there , that presentation is amazing btw. I think the first thing to consider when flatting AQ in mp vs utg is the players behind you. The button and blinds you have a propensity to be 3 bet squeeze happy. I feel like an alarm goes off on when they are faced with a raise and a call lol. So, i would keep an eye on them, you can be put in some tough spots. Also, in this spot I prefer a 3 bet pre, its should fold out the middling pairs, suited connectors and some Ax. Hopefully they have a high fold to 3 bet, if not you will get 4 bet by the top of their range which should make any easy decision for you to fold especially if there 4 bet stat is low or 0. The positional dynamic of mp 3 bet vs utg is usually a sign of strength but there is a chance they flat oop and its not the worst case, you have AQs in position and you should get a lot of x/f on the flop from that hand range. The odd time they will have a monster and slow play KK,AA and try to set up a cr on the flop and or some just flat and x-call all streets with over pairs.
In regards to what happened, I like the flop float with 2 overs and a fd. You will get a fair share of x/f on the turn from the middling pair range and if they have some big pairs like 99-KK you have some equity if you turn the A or Q. It doesn't happen though, the turn is the Tc and villain continues to barrel. Villains flop and turn aggression are indicative of strength, its sets of 88,66 and a turned set of TTs with some over pairs playing similar, i don't see villain having too many bluffs here though. Giving the fact we have blockers to AA and QQ is less likely for them to have those hands so their range is capped to mostly to sets here.
Also, i still like your turn shove, it puts a lot of pressure on that range i mentioned. You don't want to end up on the river if you decided to flat the turn with a bag of bricks and get looked up by 77,99 if you don't hit your overs. Fortunately you hit the Ace on the river and you will get a ton of crying x-calls from that range, but that's not the case, villain has the set and you are going to value town or shove town, you will be in a tough spot with a one pair hand. Well played with the shove. NH