See, and I'm just noting that in most "freeform and organic" systems, the result is that the character's mental stats don't actually matter because all the logic, exploration, decision-making, and social attributes are being roleplayed, which means they're coming from the player anyway.
Right and for a lot of people the player doing these things (thinking things through, deciding how to explore, making decisions in the situation, socially interacting with NPCS) is part of the fun.
That's fine if that's how people want to play, but if that's the case, we should just admit it, rather than pretending that we're "roleplaying." The short version is that people can't "just roleplay" a character with a mental or emotional intelligence that's higher than their own. And we've all had that player who dumps Charisma and then "roleplays" a moving speech to the King. And then the player gets pissed when you make them roll to succeed because their character has a 6 Charisma.
I reject this framing about roleplaying entirely.
There is a difference between not using an intelligence stat as your roleplay guide and not roleplaying.
Take Moldvay basic where int has two mechanical effects, how many extra languages a character knows, and whether they get bonus xp in certain classes.
I have no problem saying that a character who only knows their native language can be really smart.
You can be really into roleplaying and not care about the stats on the sheet. You can base your roleplay on a character concept.
If you want to play a Sherlock Holmes characterization concept you approach things with observations and talk through deductions and work on details. Maybe you have some arrogance. If you do not have enough to figure something out you talk about how you can't make bricks without clay.
Roleplay in my opinion is more about characterization and approach, not about success or failure at smarts or social interaction.
Aesthetically as a play experience as a player and a DM I much prefer people focus on characterization roleplay than on stat emulation. I generally want a player to play something fun for them to play and for others to interact with. I could care less if that roleplay characterization reflects stats on a sheet.
Choosing to roleplay a neutral, 8 intelligence, 15 charisma concept is fine as your goal for characterization, I just do not think it is superior to basing roleplay off of a concept of "Gimle from the lord of the rings movies" or "angry guy who is fed up with crap".
If we use them differently in game, we should probably treat them differently in the game rules. That's my point.
If you treat the mental stats as mechanics the way you do physical ones (strength gives you a +1 to hit and damage in melee, int gives you +1 language) you are not treating them differently.
I like your original suggestion of things like pick uncouth as a trait and get a benefit (inspiration, action point, xp, whatever) when you roleplay it or if it is a negative when it comes up to your detriment. A number of games do this and I think it can be a good mechanical approach if you want to encourage some roleplay from stuff on the sheet.