My main problem with betting bigger is that he bet super small on the turn, so I cannot exclude him protection betting a ten. I would definitely bet bigger on a blank river (any non broadway, non flush). If you clearly have a read that your opponents only represents a small Ace with that size on the turn, then you can try to go for a 2/3rds, 3/4th of the pot bet.

I agree that the hand is worth 1 more street for value, however how to extract this street really depends on how the ranges interact with the runout. When the turn is really +EV for our range we're supposed to bet and put our oppoent in a tough spot (this also allows us to bluff more). When the turn is not amazing, we can check more hands of this type. Our opponents are not going to start bluffing on turns that are good for our range, therefore it's better to put money in ourselves by betting than by check-calling.

Also it's more credible to represent a bluff by betting the turn and not with a bet-check-bet line on such runout, checking turns tend to really give away the strength of your actual holding in such scenarios.

I made a mistake on the river, as you mentioned their range is very strong there and I couldn't extract any more value (maybe a blocking bet is possible against weak opponents). It's true we're up against a strong range, but players are going to be happy to take a showdown with a Jx type of hand and they will not turn it into a bluff, so we can confidently fold if we face further aggression and slowplay some boats not to always check-fold.

Blocking bet would be mostly one paired 7x+ hands and few traps (2p, sets, straights). As for shove we can use my hand, some sets, and some combinations that block the most likely straights in the opponent's range.

Check shove with value hands is pretty intuitive, we have some traps and we use those. With bluffs it really depends on how thin the IP player bets there, if they are super polarized bluffing as a check-shove is not a very effective play, so you'd better off consider the other lines.

KQ is already super thin on the river, that's why I said it's a mistake to try to valuebet for my size. I'm representing boats+ so I wouldn't be happy to face a shove even if I x/raised 22 OTF. When you represent a superstrong range none tries to scare you off a pot in which you've invested most of your stack to try to fold the bottom of your range (trips, bottom boat).

I expect most of his range to be Ax when he goes for a small bet on the turn (and some Txs are also there), I can represent missed draws, while it's not very likely my oppoent will bet small with a draw, it tends to be either a top pair, a second pair or some slowplay like he had.

It's a very good flop for the BB, but we still have very strong hands, strong draws and strong hands that really benefit from protection. I would play this board as a range check, so I'm allowed to have x/raises. This hand is a perfect candidate, we have 2 overcard to top pairs (most of BB's value range is going to be 6x, 4x type of hands), backdoor straight draw to the nuts, and bdoor fd, Plus we unblock all the 3x 5x gutshots and tons of stabs with Kx Qx Jx off with a heart, which should be common stabs here. So yeah, it's a good hand to check raise, and no, we still have clean outs as sometimes even making top pair is a good out with wide ranges :)

Then be surprised :) 200x overpairs can be pretty "thin valuebets" as to be polarized we need a way stronger range (sets, etc).
I used some range I had saved there, we can add a couple t9s 98s, it's not going to change the strat as they would be mixed strats anyways and it's very few suited combinations (also we're supposed to 3bet less AQoff 200x than I put in the sim, that benefits our range).

I ran the simulation with half pot, cause while I played the hand I thought it would be the best size, I was wrong and in the hands I show I also show mistakes I made which can be pretty intuitive and common ones.

you get almost 4:1 but you're good 0% of the times. I would call also 33, hoping my opponents value owns himself with 22. My river play is a mistake there, my bet is way too thin given ranges in play.

I think the main mistake is to bet on the river there, as there are no offsuited AJ type hands to target (it would be different if the reg coldcalled from the BTN).

All the times I thought it's super bad to flat 22/33 in similar situations and ended up making similar calls they always showed up with those hands :) So I tend to assume that my opponents play bad in some previous street and they will show up with value hands that they shouldn't have (22 is too thin to jam here, but I expect a person that calls 22 to think it would be a +EV jam here).