I could see playing AQ exactly like this actually. Raised pre is standard, did a c-bet with a gutter + overcard on a board that should favor preflop raisers. The only part of my range that would be bad on this board would be the pairs TT and under. (Except for maybe 55, I might just dump that pre utg in a 6-handed game unless the game is very passive.) AQ and QT are about the only unpaired hands I would have here that "missed." I might bet 100% of my range on this sort of flop after being the preflop raiser and first to act on the flop.
On the turn the board pairs, so I could see testing villain here. This would probably be a bet-fold for me on the turn and this would be about the bottom of my range past the flop on this line. (Given QT is probably going to be able to call for a price.) I don't think villain has a ton of 5s in range calling pre (maybe A5s or 65s?) but if he tells me this story, this is the hand to let go. I don't have to defend with a broadway gutter here. If I just get flatted, I'll evaluate the river. (And again, if I hold

or

with the

and

all accounted for, it's going to be tough for me to give villain credit for spades too often, so this would be my bluff candidate, the 6 combos of AQ with one spade. Laying 3:1 on on a half pot bet, means I should have 18 value combos to bet, which would include all full houses and probably all flushes.)
FWIW, I would have preferred 2/3 - 3/4 pot sizing on all streets here. I really dislike small sizing out of position. Increasing villain fold frequency somewhat compensates for the lack of position (though "buying outs" is less important when you flop a set, of course) but I would be using bigger sizing with flopping medium strength, like Kx, as well, so I would stick with that for extreme strength as well. Or if they are going to overpay to call, you might as well go for as much as they will pay.
Me in villain's shoes? That really depends on my read on hero. I would really consider Kx about the middle of my range and I know it's a bluff catcher at this point. Pretty much everything else I get to the river with on this line is a busted draw or a made flush.
What I think is important is the read on this villain, on which we haven't been provided much detail. I tend to have a bias toward assuming most villains call too much until they prove otherwise. I expect to get paid by whatever Kx was in villain's range pre if I lead at this river as hero.
It's really hard for me to see how villain gets here on a bluff having taken no aggressive action in the hand to this point. It really feels like villain has played this to station until we got the data point that he bet following hero's check.
Btw,
@perfectjawnz, you are offering excellent responses and I appreciate how thoughtful this is.
I agree, you should definitely be betting 100% range otf here. Sizing itself gets a little dicey multiway though, so I'm probably sizing down to 1/4 or 1/3, where as heads up you can size up.
I don't like betting 2/3-3/4 for range here multiway though. If you're going that route, I think you should be a lot more polar and not be betting your whole range. Bet 2P+, and all draws, but not bad KX/JX. Sure, you have range advantage, but OOP simply means you can't bloat pot as much due to positional disadvantage. While you have all the nutted hands, you also have stuff like A3-A9hh/dd/
cc, middling KX, 55/66-TT, QJ, etc. Those hands don't want to bet big, they just want to get to showdown, so I think it's really had to maximize the entire EV of your range if you're betting large here. Maybe this is just a spot where it's very V dependent since it's multiway though. When you do bet small with stuff like AX no spades and get a fold, that's a massive win here.
for BU - If KX is middle of your range OTR, what is the bottom of your range that's called for 2 previous streets? I assume BU range looks something like 22-99, all suited connectors, some frequency of suited gappers, A2s-A9s, K9s-KTs, Q8s-QTs. OTF, you're probably folding all everything that isn't spades or a pair, so you're left with 22-99, KT-K8, QT/Q9 and XXss. OTT, after facing another bet, you're probably folding most, if not all pocket pairs except for 55, calling with KX and ss and QT/Q9. When you reach the river, you're left with straights, flushes, and KX and maybe some low frequency pocket pairs and TX. Bottom of range looks like QT, pocket pairs, and KX to me.
When Hero bets in this position, it seems like KX, QT, and pairs should be folding. If hero is checking, I think pocket pairs may be a GTO bluff, although I could be completely wrong. Maybe blocking a spade. I agree this spot is definitely underbluffed on reflection though. But I also don't think V arrives here with many hands that can call a bet except straights/flushes, but if checked to, those will bet themselves and in my experience, when people bet for value, they're much more reluctant to fold when faced with a check raise. Flushes and especially straights will probably not raise when bet to (maybe nut flush will, but I think that's a mistake - it won't be called by worse).
Compared to here, which I think is a lot more common. You have the nutflush and V (hero) has bet twice but checks on the wet card. It looks like KX and you want value. So you bet and face a raise - you're only losing to KK and JJ, but you're still beating Q9, QXss, AK, and "fuck this guy i have the nut flush, can I really fold here??" so you call it off hoping that UTG is bluffing. Plus, what other hands do you have here that can even call if you bet? straights should be folding and you don't really have boats+ here except quads.
Edit: Feel like I'm writing an essay, but I also love the discourse haha. This is my favorite part of the game.