This thread made me laugh out loud because I saw an identical proposal in 1993, and the problems remain the same. That's 26 years ago for those keeping score at home.
There are no new ideas.
First off, I don't agree with the basic premise that D&D models PCs with high INT, WIS or CHA badly.
It certainly /did/ it badly (if at all) in '93. Since 3.0, it hasn't been so bad, mechanically (OK, diplomancers were pretty horrid), in theory, if DM's'd use the mechanics, and players'd respect them when they didn't break their way...
…/IF/.
If not, well, machete, gasoline. Problem solved.
Specifically I don't agree that CHA has the same problem as INT and WIS. CHA is modeled as well in D&D as STR, DEX or CON. People lump it in here out of misguided, ill-considered symmetry.
Well, and out of the insistence that "we should just RP it" and "you can't tell me how my character feelz!"
The only time it seems to create a genuine problem is when a player dumpstats INT down to 8 or 6 or whatever and then refuses to RP it, instead acting like his character is as brilliant and knowledgeable and so on as his INT 14 or 16 previous character. But this isn't a problem with the system. This is a player refusing to RP, and no different to LG PCs who act like CN ones or a country bumpkin peasant by background who acts like Littlefinger from GoT (don't start).
It is a problem with the play dynamics of the system, with those sorts of players …which are all too common.
And, it also goes the other way, with high-mental-stat PCs, who's players took the stat only to fuel casting, and aren't up to, or don't even want to, portray the RP implications.
Whether it's a problem with the system or the players is just pointing the finger. It's easier to fix the system than fix the players, humans getting really testy when you try to pop the case and work on their non-user-serviceable hardware (your extended warranty sucks, anyway, what're you so worried about?).
But in a game with INT/WIS/CHA this would be then the DM's fault, wouldn't it? Telling a dork to "persuade that baron" in character while ignoring his stats or skills and the tell him how he failed because he didn't use his "obvious weakness for cute kittens and love of being addressed with at least three positive adjectives" is bad DMing. At least in a game where your PC's mental stats can be different from your own.
That's rather the point. /In a game without mental stats/, it's nobody's fault if the characters turn out to be no more (or less) knowledgeable, prudent, or persuasive than their players.
But as to whether mental ability scores should be removed entirely... I mean, why stop there? You could arm wrestle your players to determine whether they can lift the portcullis, or have them play ten rounds of Jenga to pick a lock. You could slip arsenic into their food over a period of weeks to see how well they resist poison.
Maybe don't do that last one.
Yeah, except for the last one.

Though you could substitute some vomit-inducing fraternity hazing trick, to resolve CON saves, I suppose.
It's a funny line, and I don't see why it's so easy to get players to cross the line that their characters are physically different, so physical tasks are resolved mechanically, but so hard for them to accept the mechanical resolution of interaction, exploration, & knowledge tasks /when they think they might do better themselves/, or when they're unhappy with an implied result.
I mean, the fighter fails to open a stuck door, they'll just curse their dice.
I can't help feeling like the endpoint of this reasoning is a game where we all just play ourselves, which isn't roleplaying at all.
It is emmersions to the nth degree, though.
But it's a game first, folks, not a thought exercise, and for that matter it is a game that disproportionately appeals to the socially awkward. You don't get to tell those players they can't play the bard if the bard is who they want to play. Sorry.
Oh, you can totally play a Bard in this variant: CHA-based arcane caster: uses POW. The end.

(What? Lore? CHA skillz? Just pick other skills. You can still take tool proficiency: musical instruments (it'll just use DEX), for instance... )