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Charisma: I really like tying this stat to magic item usage, as the packet experiments with. Make it clear to players that many magic items require a sort of personality clash, and so players with high CHA can use magic items better. There should not be a penalty for low CHA, but rather a bonus for high CHA. I like the idea of limited attunements, but that feels too much like a penalty, and many games would never see lots of magic items distributed (especially if you're low level).

Other things it could influence: mass combat situations (morale), shopkeeper prices (no checks please, just a straight +X% or -X% mod based on CHA), acquiring or retaining followers, intimidating foes in combat (fear), redirecting foes (Sanctuary-like effect), saves against charm effects, etc.

Intelligence: Languages, bonus skills, gadgetry knowledge. Improvising traps/obstacles like caltrops, or special effects like smoke bombs.

I'm sure others can think of more. I just want to see more attribute uses that exist outside of a class context.

Currently the majority of spell saves are running off DEX, CON, WIS. With four spells saving with CHA. NO spells saving with STR or INT.

 So all stats are not applying saves. So the argument that "they are all saves" is false. Im amazed so many people havnt noticed this.
Oh and closing your eyes and ears yelling "lalala there is no dump stats" doesnt make it so. Check the pregens and show me the 16 cha/int martial class.
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 But that is all off topic. The OP asks: should mental stats be balanced more within the caster classes so you cant just pump one stat to gain maximum caster benefit?

I think unless they make things like AC HP and initiative run partly on mental stats as well (higher of X or Y) casters would be left with too many stats to buy as they need to buy 3 mental stats for casting (Combt or not) and 2 physical for survivability.