In general, my problem with modeling social interactions with combat mechanics is that it ignores why we have combat mechanics in the first place.
The reason we have combat mechanics is to allow us to imagine physical combat in the game, and determine in our minds how this battle plays out and what it's outcome will be - things that would otherwise be difficult for a group to concretely and jointly imagine and agree on what had happened. We don't necessarily need combat mechanics to engage in shared imaginative play, you could come up with a different set of rules for running combat that would look more like conversation, but they are often desirable to have because they are strongly associated with the things we imagine happening in a combat - weapons are employed, combatants move about tactically, attacks are made and so forth. And they also help us have situations where we don't know the outcome. The important thing to see here is that the more granular and detailed this system, the more concretely and naturally we can jointly imagine the action. This isn't to say that more granular is always better because there are practical limits to running a system, but granularity serves a purpose and is linearly related to something valuable.
But a conversation or debate is a very different sort of thing, and attempting to run a social challenge with parallel rules to a physical challenge often runs into problems. Probably 'Dogs in the Vineyard' does it best, by doing things like making stakes customizable instead of having a 'social wound track' and by making physical raising the stakes and trumping social (at least in part). It requires a steady hand by the GM though and some general agreement to not abuse the system though. FATE or True20 on the other hand IMO does it very badly because it just doesn't seem to see why social is different than physical.
And the most important area where they are different is that we can actually play out a social situation jointly and concretely during play, so that we don't require mechanics - abstract or otherwise - to jointly imagine what is going on and what is happening. We might need some sort of mechanic from time to time to arbitrate how various statements are perceived, particularly by the imagined NPCs, but we don't actually need a detailed highly granular system like we need for combat to imagine the particular details of the conversation. The actually gritty details of this social situation are the things that are actually said by the participants, and far and away the best system we have for generating what is actually said is just to let the players imagine and say it.
And this brings up the other huge problem, which is the more granular are mechanics are for running the social combat, the less granular our imagined details of the social challenge will tend to be. Instead of more granularity adding more concretely imagined situations, more granularity makes for less concretely imagined situations. As we turn to mechanics to describe and generate events and outcomes in the social situation, what the player's actually say becomes less and less important until eventually it becomes unnecessary and redundant. In fact, ultimately we start to lose touch with what is actually said, and we instead replace the concrete with a vague narrative: "Sir John makes a cutting remark. Sir Bors is socially wounded and stutters out something that only deepens his embarrassment." Such simulation may be more realistic and true to John and Bors actually do as characters (or not!) but it's certainly counterproductive for fleshing out the scene and dramatic characterization, which will still need the concrete reality of the role played conversation.
My preferred method for social mechanics is to allow (indeed force) players to role-play. Then when I feel that the conversation has reached a sort of 'crisis point', I appeal to the mechanics to determine which direction I will take the NPC as the DM. I joint generally feel the need to impose on the players how they choose to take their conversation, because if I'm doing my job characterizing the NPC well they'll be influenced to act out anyway. It's much more satisfying to taunt a PC into combat, than it would be to roll a dice to see whether the PC is taunted into combat. It's much more satisfying to have an NPC be funny, than roll a dice and say he's funny. It's much more satisfying to have an NPC provoke emotion than it would be to tell the players this NPC is supposed to provoke a certain emotion.