This is something I've written about. Though probably not enough. There's a lot that can be said about it.
FUNDAMENTALS:
Fundamentally, there's an irreconcilable difference between social interaction and combat. Combat is force and requires no consent. The whole point of social interaction is to gain consent. Some examples of this may not resemble what would legally be considered consent. Blackmail, coercion, and so forth. But even in these cases, the aim is to get someone to do something. You're not doing it for them. The point is they're agreeing to do it, even if its under duress.
This isn't to say you can't treat the two similarly in terms of mechanics. It's just that, if you get that they are fundamentally different animals, it shouldn't be an automatic given that the aim should be to handle them the same. The unified mechanic, as it were, is a false idol. That doesn't mean you can't put together a final product that will appeal to the heathens and does keep the two types of activity similar where possible. It's just that if you want to do this well and get it right, you yourself cannot be worshiping that false idol.
REQUIREMENTS:
There is one and only one way "voluntary" or "consensual" action works. And that is all parties have to believe that they are better off doing thing than they would be by not doing thing. Being threatened in some manner can certainly sour the alternative, thereby making thing your best option. Or being deceived might make you think doing thing is the best way forward, even though in reality it isn't.
The key is understanding that how "best" is evaluated will be subjective. It's what's best according to the one whose consent is to be gained. Mind you, people do seemingly selfless things all the time. But that is perfectly consistent with preferring to live in a world where people help each other rather than living in a world where they don't.
HOW TO DO IT:
There are any number of ways to go about getting someone to do something. Some people refer to different persuasion styles. Some approaches are conscious of an entire list of things that are not anything I've mentioned. Still. Whether the person using the techniques knows it or not, it only works if it satisfies the above requirement. There are legit tons of books and businesses even pay consultants to do sales training that are seemingly oblivious to what actually gets a sale. They work almost entirely by luck. Hey, it's a numbers game, they say.
The most effective way of doing it consists of two basic parts. Find out what it is your counterpart values. And then demonstrate to them how what you're proposing gets them that. This even works in torture. How many times have we seen those movies or TV shows where the hero is being threatened, and that gets the villain nowhere. And then the villain turns the gun on the hero's friend or loved one.
GAME MECHANICS:
The two basic parts can definitely fit in the category of easier said than done. It's all about the execution. In real life, it takes practice. In an RPG, this is where stats and mechanics can come into play. Social skills should not be thought of in the vein of "make guy do stuff" quasi magical charm. This more than anything is probably where most people who pan the idea of social skills and social mechanics are put off to the idea. The prospect of being able to have meaningful, useful social skills in an RPG without it working like mind control has been something like the holy grail of social mechanics. A lofty goal, or highest ideal, but a lot of us believe it doesn't exist.
But it can. I've done it. It works. You've just got to stick to the two things social skills can do. Gather information. And present it. Gathering information can include things like doing cold reads or checks to tell if you think someone is lying. Presentation can be affected by the character's overall charisma, but it can also use specific skills, such as deception, to avoid any lies from being detected by those information gathering skills. The GM is the eyes and ears for the players. If an NPC has presented well, the GM is free to even feed the players false hunches or false social cues to simulate favorable to the NPC's position.
JUST LIKE COMBAT, JUST LIKE COMBAT:
Heathen worshippers of the unified mechanic idol, when they think social interaction should work just like combat, just like combat, for some reason are thinking of the most boring examples of combat imaginable. Like beating up on an orc, or something. I think of it more like fighting a werewolf. If you're not using a silver or magical weapon, it really just doesn't matter how good you are at fighting. You can't hurt the thing. If you have a silver dagger, though, that's a game-changer.
In social interaction, sometimes it just doesn't matter how charismatic you are, how skilled you are, none of it, if someone is reluctant to negotiate or you just don't have anything to offer. Unless you can find what former top FBI hostage negotiator Chris Voss terms, the black swan. No presentation matters, no silver tongue, unless you are able to locate the black swan through your fact-finding skills. It's like the silver dagger you need against the werewolf. Just like combat, just like combat.
GAME PREP AND STAT BLOCKS:
For the GM who wants to boost social interaction in his or her game, you have to prepare for it. We have these great monster manuals with all these combat stats for adversaries you will face. If you're going to have NPCs with whom PCs will be negotiate, you ought to have just as good notes for those social interactions. A social stat block, if you will. Just like combat, just like combat.
Note which topics or approaches to which the NPC will be inherently indifferent. Which will annoy the NPC. And where the NPC has a soft spot. Sobbing might make one hard-ass feel bad and give you and in. Someone else might view it as manipulative and react negatively. Chief Oonga Boonga might be the sort who cannot be bought or bribed and is impervious to schmoozing, but might care about the well-being of his daughter more than anything else. You need to know this. And a PC might pick up on this fact, say, upon first encountering Chief Oonga Boonga, and he's not sure if your band is dangerous. A successful social perception check, however that works in your game, might have the PC notice that the Chief moved his body ever so slightly to interpose himself between the PCs and his daughter.
PRO TIP
Feel free to leave clues to what makes NPCs tic in places other than in encounters with the NPC. One of the things that suck gigantic donkey balls about having too much social interaction in an RPG is, unlike combat, unlike combat, you can only have one person talking at a time. And so there is a real danger in a game heavy on social interaction that most players experience 20% participation, 80% watching the head of Henry Kissinger negotiate with the Brain Balls of Spheron 1. But if clues can be found through the course of regular adventuring, there's at least some great joy in watching Henry Kissinger drop the black swan that your character discovered. And you don't even need to be good at social skills to participate in this way.
Most important of all, though, is that even though I made a lot of mention about social skills, usually the best way of uncovering the information vital to persuasion is just by asking the NPC. Engaging in dialog. Roleplaying it out. That's one of the things I love about this approach. Social skills have their place. But they don't stand in entirely for roleplay. Nor do they solely determine the outcome leaving the roleplay to be naught but "flavor text." The dialog is important to the discovery and moving things forward.
NO SPECIAL RULES ARE REQUIRED TO BOOST SOCIAL INTERACTION:
I literally use what I'm describing here in every single RPG I've run. I've never needed to add stats or rules. Most RPGs already give me all the tools I need to implement this approach to social interaction. In AD&D 1E, I use the NPC Reactions table to determine first impression. Positive responses will allow players to fairly easily gather information just by asking questions, aside from things the NPC is especially guarded about (if there are any in your social stat block). Neutral reactions indicate the NPC will be willing to answer some questions, but will not be forthcoming with information unless specifically asked. Negative reactions indicate the NPC will not be willing to engage in any meaningful way.
Any time the PCs make an "ask" of the NPC--try to get the NPC to do something--I call for another reaction roll. At this point, it is no longer an "initial" reaction, so I no longer apply Charisma reaction bonus. However, the initial reaction may serve as a modifier here--neutral positive indicates that subsequent reactions should be 55% prone towards positive, for instance. This roll should be modified according to what's in it for the NPC, and how well that matches what the NPC values. The Irilian adventure from White Dwarf included a stat on how bribe-able each of the NPCs were. It's a great example of how the right notes in your stat block make social encounters work much better.
For any important NPC, any one that I anticipate will be recurring, or beginning with the second time the PCs encounter an ordinary NPC, I track loyalty per the 1E loyalty rules. Some requests may require a loyalty check. Loyalty, of course, plays a roll in whether or not the NPC will betray the PCs.
If the OP has any questions, would like me to expand on something, give advice for a particular situation, or thinks I missed something important, I'm happy to answer.