I think the biggest weakness of "social combat" systems has been pretty well-covered - they tend to make things into dull dice-rolling contests, and lack the nuance and engagement of "actual combat", because they're never as well-developed.
It doesn't have to be that way. Griftlands, a computer game, has equally developed social and physical conflict, and shows it could be done. But I've never seen a TT RPG do both well, and bolting "social combat" on to one not designed for it is likely to end in tears.
I think going with "one persuasion roll" or the like can be extremely bad too, note. D&D 5E uses an extremely random system, where even heavy investment in social stuff can mean it's only 50/50, at best, to persuade someone, and you can spend 15 minutes roleplaying and just to roll a die and you get an 11 when you needed a 12, and that sucks for everyone, even with fail-forward and so on. So I think that's a place where it makes sense to use multiple rolls, like best of 5 or something. It's not a complex/intrusive system but it means that the person who has heavy investment in social actually gets to benefit from it, in a similar way the combat-invested people and so on.
Of course the very worst is one I've encountered a bit - where a DM makes you roll after basically every sentence, and takes the approach that any failure is a total failure and conversation over and so on. That's horrific, but not as rare as I'd like to think it was. It's like if in combat, every round was save-or-die or something.
I'd agree with powers that change stuff up are pretty cool, and a good approach for D&D. Add that to "best of X" rolling approaches and you have a pretty good way to handle social stuff in D&D without bogging things down, or making things rest on one wildly random roll.
