SOCIAL INTERACTION IS NOT LIKE COMBAT. Combat is openly competitive; but the whole premise of social interaction is that you're trying to establish a basis for cooperation. If you have a competitive agenda, you need to keep it hidden. But sometimes you don't have a competitive agenda. Often neither party has a competitive agenda. The system needs to handle that.
The acid test of any social mechanic system is this: Suppose there's a social encounter between two people who genuinely do want to cooperate. Is there a significant risk that they end up not cooperating? The answer has to be "yes" or the system doesn't work. Establishing trust should be a challenge, even when the other person actually is trustworthy. That's why a combat model is the wrong place to start. In combat, if you and your opponent want to cooperate, nothing happens and you both just stand there looking at each other--there is nothing for either of you to do.
The other big challenge with social mechanic systems is that they need to accommodate player agency. The goal of social interaction is to influence the other person's behavior. But players generally take exception to being told, "The NPC rolled well on her Persuasion check, so you do what she wants." The outcome of a social interaction needs to be an incentive, not a simple "You do or you don't." If the NPC succeeds in the social encounter, you get a (game-mechanical) cookie for doing what she wants and a smack on the wrist for going against her.
Ideally, the system can also work in parallel with traditional "talking as your character" roleplaying. If you make a persuasive argument speaking in character, that should be rewarded, but your mechanical investment in social skills should also be rewarded.
So, based on all that, what should a social mechanic system look like? I don't have an answer ready to go, but I can see promising avenues. I think the best place to start is crafting the incentives, which might build on the inspiration system already in the PHB. Then figure out how to handle the "hidden agenda versus stated agenda" question, staying away from mechanics which presume that the hidden agenda is always opposed to the stated one. It's an interesting challenge.