My current Ravenloft game has a player using Mercer's blood hunter, and the PC can do several things that are magical (light his sword on fire, walk though walls, remove resistance from foes) without spells. But no one in the group would call him mundane. He's got magic blood and that lets him do crazy things.
If the fighter was constructed using the same logic, I'd have no concern with it. Magic blood, spirit, ki, the Force, or muscle power! Call it whatever you want. I just have a hard time with the idea of a local farm kid who goes off to war and comes back able to toss mountains around. Either lose the farm kid or lose the mountain tossing.
Yeah, fine with that and another reason to have a seperate class. I'd prefer if one of the origins was just "mysterious origin. you live in a fantasy world and must have been touched by something. you start manifesting incredible abilties as you level up". Some people don't want specific backstories and I see no reason they have to have it specified in a world as kitchen sick as D&D.
So, basically yes you are magically infused in some way. Done. Now you have design permission to do anything you want. Add any kind of non spellcasting magically ability. If you keep to martially oriented things then you get the mythic martial at high levels.
This gives permission for hulk/power guy, can slice an atom skill guy, extreme dexterity action hero, etc, etc.
This is level based D&D though so it even makes sense to be basically mundane at early levels and then build up to the mythic stuff, which satisfies those players that want more of a mundane feel at lowers levels.
As for the farm kid, I mean that is the oldest trope in the book right? Local farm kid finds out later he is really son of a god, the last dragon speaker, etc.