Hey Kordeth, I get where you are coming from, but I like to not think in terms of what the fighter can do as moves all the time. Fiction first and all that. Sure, on paper through the playbook moves, you can envision a situation where the wizard nerfs the fighter.
Think fiction, think cool narrated stuff, and if the moves kick in, or a custom move needs to be designed, then go for it!
In lieu of inflicting damage unarmed; you can have fists as a signature weapon, so you have the option there. Perhaps as a custom move you could allow the fighter in your game to take an additional signature weapon move as an advance, thus increasing their versatility?