I've grown to dislike ability scores that indicate personality or the intelligence of PCs, for various reasons.
I've long tried to convince people that there are fundamental differences between mental/social ability scores and physical scores that prevent them from ever being treated the same. That fundamental difference is that a person's/player's real body never intrudes into the imagined game world but that person's/player's real mind of necessity must intrude into the imagined game world, else we don't have a game to play we only have a simulation to observe.
It sounds to me from your complaint you have started wrestling with this problem and your solution you are considering is removing the mental stats entirely and just accepting that the player is always present in the character. Some games do indeed do that, and depending on the aesthetics that may be fine. But I do think that is value in the hybrid approach of traditional play.
For one, it makes certain personality choices highly punishing for some characters. If you want to play a smart or charismatic fighter, you'll be punished by being less effective at your role.
That's a game design issue. It's not really an attribute of the fundamental problem, but an attribute of there not being enough rewards for being a skillful fighter. I think we should set this aside as part of the well known "fighters can't have good stuff" problem and not a problem of mental attributes generally.
Of course, you can just roleplay your character that way regardless of stats, and I see this done constantly. Does that mean players roleplay their character "wrong?" Or is the stat wrong?
Not necessarily either one. If a player wants to play a "dumb lunk" who solves problems, he can lean into this by having his character be the guy who solves problems by not overthinking them or by accidentally hitting on solutions. It's still possible to give the impression of a character that isn't that bright but which still is "lucky" or who cuts gordian knots by simple and direct action. That's just a role-playing challenge. What happens though if you play a dumb character is the characters abilities don't help you. You won't be able to roll for knowledge, lore, insight, investigate, or whatever to get clues. Testing the character's skill will fail, so unlike the player of a Sherlock Holmes you can't lean in on your character's problem-solving ability to get around obstacles nor can you specify a character's intent and rely on their skill to implement a good plan to accomplish that intent.
There's that situation where most of the party is silent, because they're afraid of screwing up some social encounter as they side glance at the party bard. "What are you doing, barbarian? Trying to role-play in a role-playing game?! Now make a Charisma roll. That'll teach you to make the bard do all the talking!"
Unfortunately, this is not a solvable problem whether or not we have mental stats in the game. I talk about this a lot when explaining why RPGs have a combat focus and how that's just about impossible to get away from because combat has unique features shared by almost no other method of problem solving. If you do away with mental/social abilities in the game, you might actually make this problem worse, because now social interaction solely depends on player social skill and in that case why have anyone talk but the party "face" - that is whomever in the group is the most charismatic, social, and extroverted. At least when you have mental/social stats, you can use hybrid approaches and you might have different characters skilled in different social aspects - persuasion, deceit, intimidation, etc.
Then there's the trouble with role-playing characters of different intellect than your own, which (hopefully) is never really enforced anyway. "Oh, you think you just did a clever plan to stop the ogre? Well, your Int is only 8, so your character wouldn't do that! And why haven't you come up with a brilliant idea no one else has thought of yet, Gundalph?! Your character has 18 Int! Start role-playing like it!"
That would be heavy handed and incorrect, but this isn't a binary problem. There are hybrid approaches.
Should some characters just check out of the role-playing game when it's time to role-play? Should the GM keep putting the shy player with the high Charisma score on the spot?
I mean at some level, absolutely. A player who is shy but playing a high charisma character is asking for a fantasy of themselves as socially capable, so you should definitely be teasing that out of them. I always tell my introvert players to think of the situation like they are telling their character what they want them to say in terms of content, and then that character is taking their words and their intention and rephrasing them and saying them with confidence and savoir faire and as a result NPCs will react to the content of the words entirely differently. That difference is adjudicated through the social mechanics. Similarly, if you are a very charismatic person and playing a low charisma character, what comes out of your mouth won't exactly be what is coming out of the character's mouth - he'll say it wrong, he'll stutter, he'll mess it up. In both cases though I allow the player to set social "tactics" and correct tactics lead to better chances of success. The content of your message matters and will gain you bonuses and penalties based on correctly deducing what the motives and levers are in an NPC. If you can figure out what the NPC wants or what secret they are hiding, you are much more likely to succeed with a social check than if you say the wrong thing that would offend that NPC. That's where player skill comes in, and that's not only unavoidable but IMO desirable.
Should the GM berate an average intelligence player for not coming up with genius plans all the time when playing his 18 Int Wizard?
No, of course not. Having 18 Int doesn't mean you make genius plans all the time and besides which, how would that be fun or edifying?
Should he make the Int 3 cleric walk blindly into the dark room and onto the pit trap?
No, but functionally a 3 Wisdom character is blindly walking into things all the time just by failing perception checks.