I THINK Riddle of Steel does, but I'm not sure.
Those stats that I think you are thinking of are somewhat more emotional? I haven't played the game much though so I could be off.
EDIT: Actually, you might have to use a mental ability to suck up the pain when hit. Trying to recall exactly how those tables work and it escapes me.
There are number of games. A game with a lot of out of the box options is Savage Worlds, you can do all sorts of what are normally thought of as non-combat skills within a combat.
What else, umm, Burning Wheel your Perception is important for doing Access actions (to notice things around you during the fight, and to look for weaknesses in your opponent's armour and such) and your Will is the Obstacle for someone else to Intimidate you. If they succeed you have to use your Steel (mental stat) to avoid hesitating (if you fail that usually you are FUCKED). You also have to roll that Steel when you take a wound above simple bruises and scrapes.
Technically, although you don't actually use them directly, even D&D 3e/3.5e has a number of combat Feats requiring certain level of mental stats to take them. Plus a couple of Feats use your Int modifier. Come 4e you'll see a lot of Feats using the metal stat modifiers. Although that bends the "where everone is a warrior" a lot, it looks like in 4e having a fighter only party is doable. Perhaps even a mageless/clericless world as very playable?