If you equate stats on the sheet with the character concept then you will hit the disconnect. If the character concept is based on an independent narrative concept instead then it is not a big deal.I certainly hear you, but IMHO this sounds more like "play in a skilled fashion" where your PC is just basically a 'pog' that you move around to indicate where you are and to interact with some rules now and then, but the numbers on the sheet really don't match up with your RP. I guess the question is, if you are playing a PC with a low INT and you play cleverly and come up with a lot of smart solutions to puzzles that less intelligent people would not, and your ability score bonuses (or whatever for the given edition) indicate low chances of success at things that require "intellect" then it seems like your character concept is not very coherent! So this type of play does run into some problems when it comes to RP (and this is probably what the OP was thinking when he started this thread).
As a DM I want everybody to roleplay what they want from a conceptual narrative roleplay perspective (divorced from stats and classes) and to be roughly mechanically balanced for combat effectiveness.
I have no interest in trying to get players to match roleplay with class and stats. I generally do not care about what the stats on the sheet are unless there is a roll. If someone wants to play a smart charismatic Tyrion Lannister or Mr. Wednesday concept I could care less if they are a bard where the good stats for the class match up to that part of the roleplay concept or if they are a monk who mechanically is MAD for everything but Charisma and Intelligence. I actively do not want them to knock down their character's effectiveness at doing their class things just to meet a character roleplay concept by the stats on the sheet. I want the Jaimie Lannister concept knights to be awesome combatants who did not sacrifice point buy combatant stats just to meet their roleplay part of the concept of being crafty and charismatic. In a fight as part of a D&D party I want Jaimie Lannister the Knight to be as effective as Jaimie Lannister the Warlock. If a player wants to play an RDJ Sherlock Holmes expert bare-knuckle boxer concept as a monk that is just as cool a character roleplay concept to me as doing RDJ Sherlock Holmes as a wizard. Mechanically this means characters having stats to support their class mechanics.
I think 5e's background is a pretty decent mechanic for getting concept and numbers to work together as much as I want it to, particularly if the players come up with a custom background themselves. A DM throwing in conceptually appropriate proficiency bonuses or advantage supports that well too.
As far as actually roleplaying at the table I have no interest in policing my players' choices and telling them they are playing their characters wrong.
I am also fine with players with high int wizards/high wis clerics making dumb plans and high charisma characters ticking people off through the player's abrasive in-character interactions.
It is part of the whole "Tell me what you do, I will tell you what happens." aspect of the DM-Player relationship in roleplaying games for me.