I agree but think you misread my post if it seemed like I was advocating a value bet. Rather saying that bluffing seems terrible when they would never fold worse, so if somebody did bet river. Doing so for value seems
Better than hoping to fold a straight or better. Clearly checking far outweighs value Betting as they will
Check back Better hands that near us.

LHE example (to understand the concept):

CO raises, BTN folds, hero 3-bets SB with AK, BB folds, CO calls

Flop (2 players, 7.5 SB's): T32r
hero bets, CO calls

Turn (4.5 BB's): 9
hero bets, CO calls

river (6.5 BB's): T (board: T32-9-T)
hero bets...

that's how villian's calling range will look like:

ProPokerTools Hold'em Simulation

85 trials (Exhaustive)

board: T329T

Hand	Pot equity	Wins	Ties
AhKd	40.59%	30	9
((A4,A5,AK,AQ,AJ,9,3,2,44-88)!(T,99,33,22,JJ)):20%	59.41%	46	9

If we check, villian will bet with the following range:

ProPokerTools Hold'em Simulation

78 trials (Exhaustive)

board: T329T

Hand	Pot equity	Wins	Ties
AhKd	20.51%	16	0
((T,9,3,2,44-88,QJ)!(T9,TT,99,33,22)):20%	79.49%	62	0

now we can sum it up to compare betting to check/calling to check/folding...

I'm still working through this. I understand PPT to be saying that Villain's betting range is 78 hands and his calling range is 85 hands, is that right? And that Villain's bluffing range is precisely QJ?

I'm unclear how we can have 20.51% equity against his betting range if the only bluff is QJ. I interpret ((T,9,3,2,44-88,QJ)!(T9,TT,99,33,22)) to mean any pair or better that's not two pair or better before the river for your value range, plus exactly QJ for your bluffing range. So I must be misinterpreting the syntax.

What I really want to figure out is the probability that Villain bets if checked to, so that I can figure out the EV of checking.

"QJ" is 16 combos we beat.
I could include thinner bluffs like KQ or KJ, but then villian would be very unbalanced, given that we get 7.5:1 to call:

ProPokerTools Hold'em Simulation

102 trials (Exhaustive)

board: T329T

Hand	Pot equity	Wins	Ties
AhKd	39.22%	40	0
((T,9,3,2,44-88,QJ,KQ,KJ)!(T9,TT,99,33,22)):20%	60.78%	62	0

two pair+trips but not a full house and only in the top 20% of starting hands (for holdem I could just type specific hands. that would have been better) make 62 combos

OK, I realize what part of the problem is. I have no way to assess how much of Villain's range is hands that would check down, like KQ or KJ.

(For that matter, I suppose it would also help to know how much of his range is folding to our bet. It doesn't matter to whether our bet is profitable per se, but it might matter to assessing profitability of a bet versus a check.)

The real point is, we need to consider the possibility that Villain might not bet in calculating the EV of checking, since we're interested in comparing the value of betting with the value of checking, not the value of calling once he's bet.