There's a basic problem of the incentives involved when it comes to trading off prowess in your class' strength for prowess elsewhere in 5e. Ability Score increases and proficiency are extraordinarily scarce resources. Bounded accuracy means that even if you give it all she's got you are likely to still have a pretty substantial chance of failing in critical moments. You also do not gain much for diverting resources. Resources have a linear impact regardless of where they are spent.
Compare this to other RPGs which either silo off resources in a way that allows you to broaden your focus without giving up too much potency elsewhere (Apocalypse World, Pathfinder Second Edition, Exalted Third Edition) or have some form of diminishing returns on specialization so that there is a more worthwhile tradeoff between specialization and investing in your weak areas (World of Darkness, Blades in the Dark, Exalted, Rogue Trader).
Look I have tried diverting resources from my main competency in 5e. It's almost always a fool's errand. Instead of being decent in one area and deficient in others you end up mediocre at everything. It's a game that rewards specialization. There are commendable things about that, but acting like we kind of just wish that way by playing around it is not a bridge I'm willing to buy.
My overall concern is that everyone should have a chance in the spotlight, and should be able to contribute to a scene. But in my experience, that doesn't mean that everyone has to be able to competently talk to NPCs.
As an example. I have a high level Pathfinder Investigator who is phenomenal at discovering details about someone -- feats to read biographical details, legendary perception -- the works. But he is terrible at talking to people. If he tries, another party member will step in, apologize, and take over. But he contributes to scenes by feeding info to others, by searching out for info and the like.
Similarly in the FATE games I run, there are many opportunities for those without "talky" skills to find out aspects, create advantages and the like, even in heavily social scenes.
So just as I try to make sure that everyone has something they can do in a melee fight (D&D), shoot-out (Deadlands), starship battle (most sci-fi), I make sure at creation time that people can do something in a social situation -- but it doesn't have to be "talk"
I agree with this, though I’d add that it is also good, and certainly works alongside your suggestions, to allow things like a PC using a shared interest to talk Athletics or History or whatever, and use that iuPlace of Persuasion to build a rapport.
I think the goal should be that all players should be able, if they wish, to meaningfully contribute to social interaction with NPCs. Not necessarily every single instance of social interaction, but rather when taken as a whole.
I think reducing it to being good at talking to NPCs is probably going too far. It strengthens the idea that social interaction writ large is a niche that the party face fills completely and that the turnip-personality character (let's call them Gruff Turnip) is not meant or allowed to take part.
With that in mind, having different niches for social interaction, and making sure as a DM/GM/what-have-you to encourage different player characters to fill those niches, is a good way (at least IMO) to broaden the scope of social interaction. Others have already touched on this idea: the introverted nerd who opens up when discussing a topic about which they are knowledgeable and enthusiastic, the gregarious charmer, the hulking bad-cop muscle, the empathetic listener, the charlatan, that sort of thing.
So, you want to have different ways that characters can contribute to social interaction, not necessarily always in equal measure. You want to encourage the players to choose social niches that fit their characters. (Gruff Turnip could almost certainly be menacing, for instance. Or if you need someone in an interaction to be abrasive, offensive, or annoying, then Gruff is the right tool for the job.) And then you want to make a point of having situations come up in the game that reward characters from different niches contributing in social interaction.
The other thing you can do - again, as others have already touched on - is have different NPCs react differently to different player characters. The simple townsfolk are going to find the loquacious wizard boring and hard to understand, preferring perhaps the plain-spoken fighter (or maybe even Gruff Turnip, who at least has the decency to look you in the eyes and give you a firm handshake before insulting you), while of course Sage McSmartypants will have the opposite reaction.
Now, the character who has invested in social skills is a good all-round social generalist, and absolutely shines in the niches that their character best fits, but everyone else still has ways to contribute meaningfully to social interaction - if they wish. Even Gruff Turnip.
Tangent (spoilered because tangential):
This whole line of complaint about charm/domination magic makes no sense. In fact, it makes so little sense it is nonsense. It is just as much "because"-style circular reasoning as what you are complaining about.
Magic in most TTRPGs isn't just extending nonmagical things. Or, put another way, you just aren't "do[ing] something towards" a magical effect "without magic". Magic, more often than not, breaks fundamental rules of reality. You just aren't replicating fireball when you use alchemist's fire. Alchemist's fire is a fueled chemical reaction. Fireball is "manipulat[ing] thermodynamic differentials with your fingers". Constructing a glider/artificial wings is manipulating known forces to stay airborne. The fly spell is telling those forces to shut up and sit in the corner. Hiding or using camouflage is about staying out of other creatures' line of sight or being mistaken for something else. Becoming invisible magically entails letting light just... pass through you unobstructed (despite your own eyes still receiving light in sufficient quantity to see by).
Why do werewolves or vampires have only particular weaknesses (varying from system to system)? There's no sensible reason for these phenomena that you can extend logically or reasonably from nonmagical principles. If you transform, say, a human into a toad, where does all their extra body mass go? There's no sensible nonmagical phenomenon that you can work from. How do dragons exhale fire, much less lightning or sleep gas? There's no good reason - it's just magic. (Although that one faux documentary tried its best, bless it.)
Many (most?) varieties of magic in TTRPGS require overlooking physical or biological impossibilities. They are by nature inherently unreasonable. You can't sensibly justify them through any means other than "it's magic" because there is no other justification.
Suffice to say, there is no sense in complaining about the circular nature of magic, because its very impossibility precludes any other nature.
Sigh, yet another "but magic" argument that leans entirely on the idea that magic excuses itself. As a subsystem in an RPG, it's okay that your ability to engage in social encounters is at the whim of the GM, unless you use magic, which just punches the "I win" button. If we're talking game design, which, oh look, we are, the argument that magic is special and just wins is special pleading -- you're already designing a game, why does magic "win?" This is the question never looked at, instead you get this, where magic is excused for how it acts in the game design because it's magic. Well, newsflash, magic isn't real, so it has no causal power outside what it's given in our imaginations, so "because magic" for game design is the "nonsense."
Why can't my character be able to befriend someone without magic, and have that be binding on a successful use of the mechanics? I mean, this is absolute anathema in D&D (outside of 3.x, where it was extremely poorly mechanically integrated). But, it's totally overlooked if you use the Charm Person spell, because magic. But, it's not magic, it's a game, and "magic" has no causal power on game design, or how we accept the game design. And yet, every time I point this out, I get this explanation about what "magic" is (which is humorous) as if it actually has some explanatory power for why this choice was made.
I also tend to ask for rolls for manipulation type actions declarations and rely on actual roleplaying for the rest. In the case of D&D something else I do is for downtime stuff, like gathering rumors, or things like that, I'll allow any skill to be used if it applies to the group in question. If you want to mix and mingle with the hostlers, for example, I'd let you roll animal handling. I know that sounds odd, but I've always characterized it as 'talking shop', which is a very real thing. That means that every character is, at the very least, capable of mixing with NPCs who have shared skills and interests. I also like that it puts a little more emphasis sometimes on less-rolled skills and background choices.
In Agon 2nd ed - which I played for the first time yesterday - every character has a rating in Arts & Oration, Blood & Valour, Craft & Reason, and Resolve & Spirit. The default rating is d6. For a starting character, one of these four is rated at d8.
In our game yesterday we had contests across each of these domains - more in Blood & Valour or Resolve & Spirit than the others (these being the favoured domains of the two PCs), but I think at least two in Arts & Oration. These involved persuading a priest of Hera to help the PCs. One of the PCs first persuaded him to bring his Harpy-decimated flock aboard their vessel, the purpose being to aid it in passing through the storms created by the mystical Pillar of Storms and confronting the Pirate Queen. The same PC then failed in maintaining the resolve of the priest at the moment of crunch - afraid of the storm, he instead led the PCs' vessel astray, leaving them vulnerable to an assault by the Pirate Queen who had herself taken control of the Pillar of Storms.
