Ok, Then I read your position correctly. There's nothing necessarily wrong with allowing creative players to use their best stat for everything (or a lot of things), but it's not my cup of tea, and I wouldn't like it to be the 5.5 RAW for intimidation or anything else.
That is ok also but I would prefer to make abilities more relevant and to give characters that not have prof (intimidate) a chance. Also why it is fair if you change the DC of the intimidate check to match the fiction does all dms do that? are they encouraged to do it?
I also think that it is conceptually easier to switch the Ability check than to adjust the DC.
Easier? Sure, but I believe some interesting granularity is lost in the process. I'll try to explain what I mean by going back to a previous example.
Example: A strong PC with low Charisma tries to intimidate some thugs by bending a metal object with their bare hands.
If you allow the PC to use their 20 Strength instead of their 8 Charisma, well, that's just it. The PC gets an hefty bonus to the roll compared to the default of using Charisma.
What about the approach I suggested, on the other hand? As a DM, I'm privy to information that the PC isn't. What if the "thugs" are actually polymorphed demons? They might not be impressed by the mortal PC's feat of brute force, right? Thus, I would set the DC high, and ask for a Charisma (Intimidation) check.
If, however, the thugs are actually a craven bunch of nobodies, I would still ask for a Charisma (Intimidation) check, but I'd set up a fairly easy DC.
But one important thing is, regardless of the circumstances,
the player's conscious choice of assigning a specific score to Charisma would not be bypassed or trivialized.
A high Strength character with a nice 14 Charisma would be better at this intimidation tactic than an high Strength character with a low 8 Charisma. Which I think is good and fair! Brutality backed by a strong charisma
should be comparatively more effective, IMO.