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This is why Intimidation is a floating skill at my table
(startrek.website)
Humor, jokes, memes about TTRPGs
There are a lot of things I like about 5e, but charisma making you good at ALL forms of charisma simultaneously is one of my least favorite changes they made.
Most of the times we decide spontaneously what ability to use for a certain skill. The fixed stat+skill is super annoying and breaks immersion.
The wisdom 20 / int 8 Druid not being good in medicine? ... yeah maybe not good in school-medicine but knowing what herb can treat what illness is a thing of wisdom, not intelligence by default.
Then, yes, Strenght for intimidation.
Intelligence for deception - think of an elaborate network of pseudo-facts and weave them together in a complex way so the "opponent" is so overwhelmed that he just choses to believe you.
And so on so on...
I like your philosophy of trying to pick a more appropriate skill when it feels right. I need to remember that one for my next session.
I mean it's literally the way the rules intend it. Most just get confused by the character sheet showing the most common ones.
I'm playing for 5 years with my group now and still get some confused looks when I ask for e.g. a Charisma (Investigation) check.