DO3SNOTCOMPUTE...

I personally don't understand how raising half of your stack with the intention of folding to a jam ever makes sense, in any scenario. I can see doing it instead of jamming in spots where you are never folding, just to use max timebank in order to increase probability of making a pay jump in certain situations, but if you are planning to fold to 3bets, why ever risk half of your stack- especially when you only have 8-12bb?

While it is bizzare, to think that it should NEVER be done is just a premise to missing out potential value. While I've seen analysis of FTs where some top pros do this, there is no indepth strategy discussion. If I have to theorize though I would say that it most likely applies to situations where the bigger stack(s) are so far ahead of you that chances of shipping are low. So ICM is our main source of making money, through payjumps. and you are possibly going to run into a spot where you can likely win a payjump if multiple stacks go in,and you fold to lose a part of your winning chances and it's okay cause it was low anyway, but gain a substantial $ev. Basically, losing 50% of a 100$EV(shipping EV) might be okay if you gain 25% of a 500$EV(payjump EV). Of course, getting those exact numbers is a completely different ballgame, but that's the idea.

Yeah I just don't understand what the benefit of raise/folding 4bb of an 8bb stack is when you could instead just raise/fold 2.12bb. I guess if the rejamming ranges of opponents change a ton facing a 4bb open compared to a 2.12bb that would be a big factor. Guess I would have to do some theoretical analysis and make a lot of assumptions to see where it could potentially make sense. I think you are correct that this would only apply in ICM scenarios, as doing so in a cEV or close to cEV spot seems like it is just burning money.

If top pros are doing it, there has to be some merit to it. Has to be based on a large increase in fold equity for the larger size.

The advantage is that if there is not a major collision going on, with the 4x open you're denying fold equity to aggressive rejams unless you want a naked AA in your range which can't be a good idea. Also, you don't want the BB to flat cheap vs 2.12bb, since the aftermath of this is often playing for stacks especially given the low SPR , or just being lord of the Ev losing NITs.
The 4x accomplishes the same as a 7.9x (keeping fold possiblities in extreme situations), so why not save more in that necessary evil situation.

4x might be the lowest sizing that wont be defended versus if we must answer why the 4x benchmark and not others. Although no one will party if you make it 3.5x either.