I think I would just fold turn vs a PSB. You hold Kc blocking some bluffs villain may have. I don't think villain needs flush or 7x to shove this river as you'll have very few flushes in your range with the Ac being on the flop. I just don't see players turn 88-QQ here into bluffs very often. I would focus more on pattern recognition and less on exact ranges. Is this a spot that is usually bluffed? Being call 3bet and bet pot on the turn?

Turn ($0.53) 7 4 A 8 MP bets $0.51, Hero calls $0.51

why didnt you cbet 30% OTF?

Agree with dm0sky here. We could cbet range here (and should if we plan to bluff here basically always).
RunItTw1ce: I think checking back here makes sense if ranges are whider. We simply have not enough weak A i guess to balance that spot in these positions.
If we 3bet 88+/AQ+ and maybe A5s from time to time, we check back mostly PPs.
Its also way more easy to cbet 1/3 (maybe even 1/4) OTF/OTT and fold to a raise or Xback river. We might also get called twice from TT/JJ.

I agree with RaoulFlush that we should range bet flop.

Unless we’re checking everything, then our check back range is horribly capped and face up (due to the lack of weaker Ax in our range we can check back to balance our checking range).

We’re in position and have a pretty substantial range advantage on a dry A high board, so range betting should definitely be permissible. It also simplifies things: it means we don’t face huge barrels on the turn, which we’ll then struggle to defend against with our horribly capped check back range.

As played, I think I’m probably folding the turn. You block some of the very few bluff combos villain can have with the Kc in your hand and I don’t agree with OP that villain will be betting hands like QQ here. They have stable showdown value but can’t expect to get called by too many worse hands. I think they have very little reason to want to bet here.

I’m definitely folding river. I just don’t see villain having enough bluffs, while he has plenty of Ax hands which defend to the 3bet pre.

I think the big mistake in this hand is checking back flop: it complicates things and leads to the tough decisions we face on later streets.

I've seen Pio check back or mix high frequency check back here with strong hands like AK and AQ as well. It's still relatively easy to get money in on two streets. I think A lot of AJ and AT hands should check back as well as KK-JJ here. Then bet a lot of your AQ, AK, and air balls that missed.

Here I loaded the hand into Snowie. I checked the 1/4th pot, 1/2 pot, pot, and 2x pot sizings. KK is checking at 100% for higher EV. Cbet range should be more polar IP in this spot imo. Can't just apply 1/3 as solution every time. Yes, its +EV, but its not the highest EV.

What happens when you 1/3 flop, check back turn, and face a river bet? Comments above said he can call flop still with QQ and JJ, so given you show weakness, I think QQ and JJ can be turned into bluff on river.

Here is the turn action. I had to use 10/20c since Snowie does not go lower, but BB is still same. Just did pot on the turn, which is only off by 1bb, but you can see it's clearly a blunder. If you check back enough Ax here on the flop you'll have much better hands to choose from, making KK is relatively easy fold.