I ran the spot with very wide limped ranges. With similar sizings and you were right, GTO is not concerned with meeting MDF here.
In my sim: OTR we bet 1/3rd with any pair+. And this is the range explorer to vizualize checking range: https://gyazo.com/ea3d39783cd78bc7d81e0326d62d45d2.

After checking this river PIO only defends 30% of the time vs 3/4th bet sizing. Could help me understand the theory of this spot? And why we don't use naive bluff-catching freq(1-A) here?

Generally speaking, MDF is what you want to aim for on the river...but it's based on sort of a toy game where 0 equity bluffs are always in villain's range. What if villain's weakest hands (as determined by previous street decisions) overlap with some of our potential bluffcatchers? In order to hit the MDF in those cases, we freeroll ourselves on those bluff catches and lose EV by calling.

What is MDF?

Kostanzhoglo
MDF = Minimum Defense Frequeny and it refers to the calling frequency we need to have to make our opponent's bluffs indifferent. It is also tagged (1-a) because this formula is used to calculate the theoretical MDF.

In the formula MFD = 1-a you have | a = opponent's bet / (pot + opponent's bet)
So our MFD = 1 - opponent's bet / (pot + opponent's bet)
For a half pot bet this is = 1 - (0.5/1.5) = 1 - 0.333 = 0.666 = 66,67%

I can recommend you these two videos from the bottom of my heart best watched in this order if you want to become solid in this concept.

Improving on "1-A" (Steve Paul) (Introduction followed by Theory)
1-Alpha (Tyler Forrester) (Mainly introducing Blockers to the '1-a' concept after recapping the Theory)

I initially thought SB can 2barrel small all A-highs ott, actually just some best A-highs at equilibrium (and BB doesn`t do much turn raising) but even this case BB calls turn w/ all good Q-highs and most good J,T-highs.
Betting small yourself 3rd time is still ~ +0.5bb better than checking.

Funny thing is spade makes it a worse bluffcatcher but its exactly spade which makes it best and highest freq 2barrel hand (among combos of similar rank) and implies most of your 7x will have a spade otr.

You say (in game):

I think its river situation which is relatively unimportant to defend accurately.

above:

What if villain's weakest hands (as determined by previous street decisions) overlap with some of our potential bluffcatchers?

Doesn`t it mostly relates to your ~0 ev bluffcatchers in equilibrium (which becomes -ev when opponent starts to turn SDV in a bluff) and decent +ev catchers (which beat that SDV) should still call?

Ive got you found yourself in a vague spot but it looks gross as played. Looks like you underrep to hero fold later bc you dont care about defending hands which are less than a boat.... for some uncertain reason :)

Nice video.

In game I don't think I accounted for enough Ax in my turn range, nor did I realize that 7x is probably a good river block bet after getting flatted twice here. I've been reviewing the spot a bit and I'm about 80% sure that I thought I was playing a 3bet pot (lol), which gave me an overly tight perception of LL's range at turn/river.

@SMS I think with that in mind your reenactment of my in game thought process is the most accurate post so far! I just want to point out that my comments about not hitting MDF and the overlap between bluffs and bluff catchers is not intended to conflict with GTO sim results. If pio shows hands are 0ev or higher as river bluffcatching hands, I don't mean to imply we should be folding more of these than suggested. I'm only offering an explanation for why in some situations we won't have enough 0ev+ calls to reach MDF from the above formula.

tl;dr i should block bet river and LL might have more bluffs than I first implied

If we can bet bigger with a big enough % of our value range, and therefore don't have to reduce our betting frequency too much (or possibly not at all as we add bluffs), our range's EV should increase. Why are you hesitant to bet larger here? The turn is one of the best ones possible for my range and villain will rarely improve...I think 66 is plenty strong enough to overbet here.

21:30 - I had planned only to use 1 bet size here, but I think it's almost definitely higher ev to create two ranges here as you described

27:30 - I think the removal effect that I touched on in the video is pretty important here, and the main reason I don't mind sizing small with my Kx. We can't size very big here for 3 streets even with Kx because we'll allow LL to fold so many of his non Kx combos. I think paired and double paired boards end up breaking a lot of the intuitive rules for big bets / overbetting, and I also like to toy around with less polar ranges on these boards so that we can have a less transparent strategy.

40:30 - I don't think he should do this often, but with an approach that uses small turn probes, it's not an awful choice. He needs more protection with non-club Qx and so he's denying equity to some really poor hands in LL's range like two undercards with 1 club.