Currently the way character classes are is rather ad-hoc. There is no desire to balance the classes between the stats which purely in my opinion is a mistake.
Just as a warning, this experiment has historically not gone well, because you end up redefining what stats mean and then everyone is confused why your Charisma replaces Strength when swinging a sword and the game is too silly to play. It's really
hard to make Constitution a primary stat, because you can't reasonably attack anyone through it. The same was true of Charisma, before they redefined it to include innate magical power.
If you're familiar with the recently-defunct Diablo-clone known as Marvel Heroes, they attempted something very similar to this in their dying months. Leading up to that point, stat balance had gotten so bad that the only stats that anyone cared about were Strength, Energy, and Fighting; because those were the ones that affected your damage. Characters who punched or used most weapons (Juggernaut, Deadpool, Mister Fantastic) stacked Strength and Fighting because it let them hit harder. Characters who had powers, or used some weird weapons (Cyclops, Star-Lord, Doctor Strange) stacked Energy and Fighting. Durability, Intelligence, and Speed were all pretty much meaningless.
Their solution was to give every hero exactly two stats that they scaled from. For example, Mister Fantastic was defined as an Intelligence and Durability hero, so every point in either Intelligence or Durability gave him a bonus to damage; and every point in Strength or Fighting gave him effectively nothing. Even if you made him as strong as The Hulk, or as skilled as Elektra, he would never punch any harder. It was really bad, and the game didn't survive much after that.
I'm not saying that you're necessarily advocating for such an extreme, but that's the road which this line of thinking has traditionally ended up going down.
As you level up you gain major abilities that work off your primary stat and lesser but useful abilities that work off your secondary stat.
I think this is a great idea. Instead of saying that some characters can punch with their Strength and other characters can punch with their Dexterity or their Charisma, all stats should work the same for every class, but different classes should want different stats due to their specific combinations of abilities. "Everyone who cares about swinging a sword should care about Strength, at least a little bit," seems like a great place to start. But then you can distinguish classes by what
else they can do.
If paladins care about both Strength and Wisdom, then it should be because they both swing a sword
and cast divine spells. If they care about Charisma, then maybe it's because it augments their leadership abilities in some ways (such as through a special aura feature). If they don't particularly care about Intelligence or Dexterity, then it's because they don't have any class features which really rely on them.
