Sorry, this post is to intellectual for me, I don't understand.
Let me try to help:
RPGs are about conflict and it's resolution. "Can my character convince the king to support our mission," is a conflict. The game is, for this moment, going to be about how this conflict resolves.
When we do this, we can do it in a bunch of ways. The way that D&D generally proposes is "ask your GM what happens." The GM can call for a die roll or just say. If they call for a roll, the GM sets the needed roll and also still says what happens after the roll. So, in this kind of play, the GM has all of the authority to say what happens and can use a number of ways to do it. The default assumption here is that the GM is honest, and has the good of the game in mind, so if this is so the GM is acting to best suit the game. This, however, requires a huge amount of trust in the GM to be this person and to have the skill to not make (many) mistakes of either resolution or understanding of what the player is trying to do. Let's assume that in general this is so -- the GM is competent and trustworthy.
But what if a mistake is made? What if the GM, in a moment of being a human, puts their thumb on the scale? What if the player misunderstand the GM's perfectly fine ruling as either of the above (another moment of being human)? This approach depends on trust, and trust is very, very easy to damage. Still, in general, people can usually trust their friends, and a good GM can earn trust and even repair it if something like this happens, and also we're often conditioned by life to deal with less than perfect trust relationships so, in general, this approach works well enough. However, there's always this specter of lost trust to this approach -- lose that trust and the whole thing crumbles. This is the exact thing that most of the RPG horror stories revolve around -- disregard, loss, or abuse of this trust between GM and players.
So, then, what do robust mechanics do to this? They remove quite a lot of that specter of lost trust. If there's an open, understood way to resolve RPG conflicts and the inputs are clear and the output space is also clear, then there's no where that trust can be lost because trust isn't in the system. This is that competitive integrity
@Manbearcat is talking about -- it's a clear resolution system where any abuse it immediately obvious to all participants.
Now, to hopefully head it off, this is and isn't about trust. It's not a matter of one not trusting their GM so much as it is one of having a level playing field. I'm a GM for systems that don't have any kind of structure in place for social conflicts (5e) and for systems that have robust structures in place for social conflicts (Blades in the Dark, for one). I certainly don't like the social structures of Blades because I don't trust myself as a GM! What I've actually found is that I'm more free to bring heavy consequences and direct pressure under the Blades system because I do not ever have to worry about any of my actions damaging or calling into question trust.
Finally, I've used the term "conflict" throughout for a reason. It's not about "winning" but resolving conflict. If the players do not want conflicts resolved in favor of their characters, there's a larger discussion that needs to take place. Social mechanics are not about skipping to the "win" but about how you resolve the conflict. Often, robust mechanics are not at all single rolls, sometimes they're entire sessions. In Blades, one of the score types is "social" and that's a session's worth of play to resolve the question or ask that kicks it off. These results are quite often messy in the sense that they're not clean and easy for the characters but full of not quite what you wanted paired with unforeseen consequences. So any quick quip of "social mechanics are just button mashes to 'win'" is a category error.