Nebulous
Legend
13th Age does like this but takes it another step - weapons (and armor) are just categories and you can define the specifics of your character's weapon. But it also breaks up the mechanical bits by class. Fighting types do the most damage with a weapon, casters with the same weapon will do less. Rogues go more with light weapons than anyone else, even before sneak damage.
I would love to port this over to 5e, but i think if it's not outlined as "optional" in the DMG my players would be less keen on trying it.