To recap the hand,
Preflop: UTG Limps, Hero raises to $60 in EP, MP calls (1250), HJ calls, CO calls, UTG calls
Flop (315) Qd Jd 9c
Checks to Hero, Hero checks, MP bets 165, folds to Hero who calls.
Turn (645) 4c
Hero checks, MP bets 325, Hero calls
River (1295) 3c
Hero leads 400, MP calls
Hero shows AdAx, MP mucks
In a multiway raised pot, QJ9 with two diamonds smashes the calling ranges of the hero's opponents. For most players, many two pair and straight combos, as well as all sets are in their range. They could also have a hand that is currently behind with a ton of equity against aces like J10s or 10x of diamonds. If John were to bet and get raised he would have to fold. With three people left to act, by checking John is able to see what the three people behind him choose to do. If someone bets and anyone else calls or raises, I think John also has a pretty easy fold. But if just one person bets and everyone else folds, John can then call and feel pretty good about how pocket aces is doing against the bettor's range of hands. On the flop check-deciding pocket aces seems like the only viable option.
When Pam, the tightest player in the hand bets, I don't think John can feel great about his hand. On the podcast we came to the conclusion that she would value bet as weak as KQ, and also bet her strong draws in this spot.
This range is a slight equity favorite against John. Now a mistake a lot of people might make is that because Pam's range is a slight favorite against theirs and they are out of position and not sure of what to do on future streets, they will fold to this single bet. But this would be a big mistake because given Pam only bet half pot, once it folds back to the hero he only needs 25% equity against her range for this call to be at least breakeven. In my relatively tight ranging of Pam John has 49% equity so no matter how one ranges Pam, I think it's a pretty clear check call.
The turn brings a complete blank. Against the range that we assigned to Pam on the flop, the equities actually barely change on this card. John goes from having 49% equity to being a 51% favorite. On the podcast we never discussed leading out on the turn, and thinking about it more I believe it does have some merits. If Pam is a really tight player who will only raise straights, betting out allows us to get max value from the draws and worse pairs in her range that would check back. When we get called and are behind, I think betting the turn might make Pam scared of a nutted hand checking the river to her, especially since the hero was the preflop raiser. This could allow John to get max value from all worse hands and lose the minimum against her better hands, because he would of course fold a to a turn raise.
All of this being said, I don't feel like with the information I have I can be sure that Pam won't raise her sets. If she raises her sets and some two pair combos, this bet becomes really bad. So I agree with John's check, but I think there are tight opponents like John described that would only raise the turn with a straight where leading out would be the best play.
Once Pam bets, I think we can take out most of her combinations of pair + draws, flush draws, and straight draws (TT).
Against this range John has 44% equity. But given Pam's bet sizing of 325 into 625 and only needing about 31% equity, John has a very clear call.
The river brings a 3 of clubs, completing a backdoor flush. Made flushes on the river are a very small percentage of both player's ranges so this card is basically a blank, although it might seem like a scare card at first. Against Pam's betting range on the turn, this card barely changes the equities: John now has 46% equity against her range. Going with John's assumptions about Pam's postflop tendencies, I agree that it is very unlikely Pam would triple barrel in this spot with a pair + draw or flush draw hand (the nut flush draw John is actually blocking). So if John checks, I think he has to fold to most bet sizings as it's hard to imagine Pam as described would bet KQ or AQ.
In game, John opted to bet 400 into 1275 with another 400 behind effectively. I think all of Pam's queens will call here, and only rarely does she bet a Jack or worse pair on the turn. So it looks like Pam will be calling with all of her one pair hands on this river card for a 1/3 pot bet. How many hands in Pam's range are just one pair? In my estimate 46% of them. So about half the time by betting 400 here John gets another 400 in value he wouldn't have gotten by checking the river. But he also loses another 400 the other half of the time when Pam calls with a better hand or raises him.
If John was confident in his read that Pam almost always has two pair+ when she bets this river card, there isn't that much of an EV different in betting a sizing that would get called by a one pair hand versus check folding to almost all bet sizings. But Pam didn't have exactly 50% one pair and 50% two pair+, it was 46% one pair hands which means John is losing a little bit of EV by betting.