We've had a bit of a problem recently with players wandering away from the table outside combat because they feel they have nothing for their characters to do. The problem is that the one character with the best social skills in the game, is also played by the player who is the best skilled at role-play, and so nobody else feels they need to do anything. If the GM tries to force another player into the scene, for example by having an NPC address them directly, they end up looking like a deer in the headlights and then complaining that the GM is trying to hurt the players by targeting a character's/player's weaknesses. Is there a way around this?
tl;dr: play the game you'd like them to get better at, possibly embedded into your ongoing game.
Your players don't enjoy social scenes.
Right now you're trying to make some players play a game that they don't enjoy. Not only are those players not asking you for help learning this game, they're actively complaining when you try to engage them in that game.
Carrots and sticks within the 5e system just seem like hey're going to exacterbate the problem you're having. There are clearly things about 5e that they enjoy, or they'd not be coming to the table; there are things about 5e they don't enjoy, and they've told you as much. How much can I push my players in a direction they don't like before they abandon my table? isn't really a meta-game I hope you want to play.
Your system is not helping you.
5e--and D&D generally--tries to be a lot of different games simultaneously. The downside of this is that a player who really enjoys game A has to share session-time with Games B, C, D, .... Luckily, most of us enjoy many of the sub-games, but there are always some that don't really suit our fancy. And we tolerate those as the cost of playing the rest.
But in your case you've got all-players-but-one not enjoying game C, let's call it.1 So game C simply exists at your table as a cost I must pay to play games I like.
This, then, is the crux of your problem, I believe: the game you want them to play is actively hindering efforts to play the game they want to play.
Not all systems are like this.
This may sound crazy, but because game C obstructs other games they like while in 5e, I think the solution is to play game C on its own.
Have you ever noticed an Attack Wing complain about there being too much combat? No, because that's the game they signed on to play. Ever noticed a chess player complain about having to evaluate the relative merits of different moves? Or an improv actor complain about having to speak in character?
There are role-playing systems out there that don't have a combat mechanic, that don't really have a sense of character-building.2 Ones that only have social-interaction scenes as part of their gameplay. Playing a game like this (a) removes the sense of this is keeping me from this other game I'd rather be playing, (b) usually includes structures for bringing players along to avoid the deer in the headlights feel you're noticing, (c) doesn't up the ante by putting a player in a if I screw up this part of the game I don't like I'll make things harder in the part of the game I do like situation, and (d) can3 level the "playing field" between your one player and the rest.
"But we like playing our D&D game!"
So do I =)
So let's make another 180\$^\circ\$: make this a part of your D&D game. My example: when starting a new campaign, or adventure, or even just when entering a new part of the world in an ongoing campaign I will often pull out Microscope and spend an hour or two exploring the history of this new region. The pitch is: "hey, all, I haven't thought much about this area yet, and I also know that if we only ever use my ideas then we're going to see a lot of repetitiveness. Can you help me figure out what this place is about?
(a) we're putting away the hack-and-slash game for an hour by GM fiat; (b) we have a social mechanic that handles sharing spotlight/narrative agency; (c) the players now have an opportunity to set things in place that will advantage their characters; and (d) that particular system is really good about explicitly sharing narrative "power" among players.
And now you'll be able to play one more of the D&D games better =)
1 - "C" is for "chat."
2 - I mean "character building" in the typical D&D sense, of poring over handbooks looking at options and trying to discover mechanical and roleplay synergies, in whatever mix one fancies.
3 - a game build on social scenes will often feature spotlight-management and -sharing mechanics that simply don't exist in the free-form social environment of D&D.
If, when you ask other players to participate, you're requiring rolls against social skills, then they have a valid complaint: they should leave the talking to the Face, and stay out of it.
If you really want to involve other characters in the role-playing scenes, don't make them roll against the skills they're bad at. Instead, you should grant advantage to the Face's rolls due to the assistance and participation of the not-particularly-social characters. This way, the non-social characters' poor skills aren't dragging down the characters' ability to get things done. Also, make sure the players know how you're going to handle this. If you decide to add this "assistance = advantage" mechanic, but don't tell the players, they might still think that you're working against them.
Finally, roll less. The majority of conversations require no roll at all. For those where rolling is required, just one or two rolls should cover the entire conversation, taken at the point where the NPC is making a decision.
Talk to the players about this. Tell them that you'd like them to participate in the role-playing scenes, and why, and how you're going to handle the mechanics.
Work with each player to make sure that their characters are fleshed-out, with a background and personality. Make sure that both the players and their characters care about your plots.
As the players start to role-play, don't punish them if it turns out badly the first few times. They'll just learn that role-playing is the Face's job and back out again. By punishing them, I mean let them succeed at what they're trying to do – a blundered interaction may look like an "interesting development" to you as the GM, but to them it will just look like everything went wrong and they shouldn't try this again.
High Charisma doesn't mean everybody likes you
This is one of the major misconceptions that, to be fair, rules-as-written seems to exacerbate. "I have a high Charisma, therefore everybody likes me." When this is how the game world works, it makes perfect sense that the PCs put their Face character in front and let them do all the talking - it's strategically sound.
However, this simply doesn't take into account the variables of character, personality and circumstance that make a pen-and-paper collaborative RPG so compelling. Instead, go by the rule "High Charisma doesn't mean everybody likes me."
Mechanically, advantage/disadvantage is perfect to facilitate this. Depending on the personality, nature or pre-existing history with an NPC, grant advantage to the PC who would be a good fit and disadvantage to a PC that the NPC absolutely despises.
Let's say your characters are trying to negotiate with a tough-as-nails, jaded town Sheriff.
Your high Charisma Face character steps forward - an impeccably clean, snooty noble-born Sorcerer. The Sheriff's eyes darken, judging the Sorcerer's fancy clothes and pompous attitude. "What do you want, mister? Or should I be calling you 'my lord'?" He sneers and spits. (disadvantage)
Or perhaps the battle-scarred Fighter steps forward. "I respect a military man - how can I help you, sergeant?" (advantage)
Or the young female Rogue. "My god, but you're the spitting image of my long-lost daughter." He tears up. "What can I do for you, little lady?" (instant success)
Or the elf babarian. "I don't want nothin' to do with your kind - kindly be about your business and leave." (instant failure)
Given a high Charisma and good social skill proficiencies, this method allows the Face to breeze through social encounters with a compatible NPC (advantage), while still allowing the Face to win the charms/attention of a hostile NPC if they're skilful enough (disadvantage).
At the same time, characters with a low Charisma can still be very valuable in situations where the party has to deal with particular NPCs who might simply be warmer to certain kinds of people/races/personality types.
For your players, this will break the "socialise by number crunching" aspect of the game: to be strategically sound, they'll need to roleplay to learn the preferences of an NPC before committing to a single strategy.
(As a side note, this also allows more depth and interest in the NPCs themselves - they're not all robots who can be reprogrammed by a high charisma character in the space of a single die-roll - they're their own characters with discrete motivations and preferences. There are some people they'll tolerate reluctantly, some people they're happy to engage with, and some they won't listen to for a second.)