That is what bothers me...

That was my main point for checking QQ instead of bet/folding it.

It might be fine but i just do not think bet/fold like half of our range is good
(AJ,AQ,88,TT,JJ,QQ).

We still can have the best hand you know.

Very valid point. However, I think he just has too few value combinations in his range. I count ~40 combos (K9/97/74s/44/77/99/K7s/K4s/some other Kx combos not all of which c/raise the turn), and I'm guesstimating that he's risking 3BB to win 7BB, so he can bluff with maybe 18 additional combos. (hope someone can check the math).

But he has every straight combo (65/86/T8/JT : 60 combos) that all have 8 outs, many flush combos (~15+ combos), and all kinds of pairs.

Maybe bet/folding half our range, or bet/calling with some of our range, isn't nearly as bad as giving 100+ combos of 5 -15 outers a free look at the river.

Not always imo.
When you take a line to bet/fold you better be sure to be right because losing the pot is very costly when you have the best hand.

We often use the principle of : better lose a bet by betting the turn than lose the pot by failing to bet and giving a free card that cost you the pot at the end.

But imo we use this principle when we actually have to call down.
Not as an excuse to bet the turn and fold like half our range ( unless of course you have solid reads vs your opponents).

I mean if its bad of failing to bet the turn and losing the pot on the river due to a free card, than it must be aweful to bet and folding wrongly , giving not only the pot but a bet as well on top of that.
Bet/folding half your range would be very exploitable and you could suffer from it.

Now about how many outs your opponent have, yes it sucks but the pot is 11,5 sb and you need to bet 2sb on the turn.
Your opponent needs around 9outs minimum out to have 2 sb value in the pot .
Giving a free card to have a 5 outers sucks but is it worth it to risk 2sb to protect from a hand that has 5/6 outs agaisnt us ( which is worth only like 1.2sb for our opponent) when we have to fold it ?

If it was a small portion of our range with QQ , i would not have a problem to bet/folding it but here , we have so many other hands that would benefit more using this line of bet/fold , i would rather use QQ with another line to play with it.

And only 2 other line is present , to bet/call and play poker on the river with a hand hard to play because it is midling strengh hand once we get raised or to check and have a pretty good hand on the river with the line we took ( yes it is easy , we have a very strong bluff catcher hand so easy call down ).

I favored the second option because having a good hand on the river by checking makes it much easier for me to play at the end.
In decent size pot, i like to play hand where i minimise my mistake while bet/calling here with QQ on the turn would put me in a hard spot on the river.

This is a bit oversimplified.

Say there are 6BB on the river, you bet and villain raises. Pot will be laying you 9:1 on a call. You don't need 100% certainty that villain's hand beat yours to profitably follow through on your b/f. If villain bluffs 5% of the time, with exactly the nut low (guaranteeing you beat his hand), you should fold all the time, even though he will win the pot with the worst hand that 5%. This is you exploiting his low bluffing frequency, specifically you saving a bet that 95% of the time.

To be clear, b/f is only made worse on the turn when our opponent bluffs disproportionately often, but here we assume he's playing GTO.

If you were to bet the turn, the pot would be 6.75BB. He stands to win that, plus 1 or 2 BB on the river. Lets say on average he wins an additional 1BB (sometimes you check behind river, sometimes he c/raises and we fold, or he donks and we call, etc.) So getting 7.75BB:1, he only needs 11.14% equity, which is roughly 5.5 outs.

So I think betting to extract value has more need than you're crediting, as he has the equity to continue with nearly his whole range.

I think the real question here is: How wide of a b/f and turn-check range should we have here, if any, and what's the logic behind constructing it?

With that said, I'm inclined to agree with you. If we need to be GTO, we can't bet 100% of our range on the turn, as then our opponent can exploit by raising more. If that's the case, putting QQ into our turn check range and 88 into our b/f range seems correct.

When an average player check raises 2 players here oop, it is a standard fold. Against a very good player, a 3 to knockout the button and hopefully take a fee showdown unimproved on the river is very reasonable.

I could never check my hand here unless I was 100% committed to folding when it came back bet even once. In this hand I couldn't do that.