Now, I shoot more squared than I do bladed and I tuck my elbows in tight with long guns. The only thing I do different with a shotgun is push forward with my firing hand.
The only reason the stock even touches my shoulder/chest is to stabilize the rifle. So, my shoulder is not absorbing the recoil nearly as much as it could be if I pulled tighter.
I don't spend much time on shotguns. On average, I shoot about 750 rounds a week. Maybe 10 of that (if averaged per week) is through a shotgun. So I'm certainly not an expert on them, but I have trained on them in different environments and have put over 200 through one in one day with no soreness of any kind.
What I think I'm trying to say here is that the same result can be had with either grip, but maybe people just use more of a push/pull technique after installing a pistol grip...
I don't know.