How about divorcing weapons from damage die size entirely, and let it depend on skill/proficiency?
with
- An arming sword could be main-handed/slashing/defensive
- A maul could be 2-handed/crushing/brutal(2)
- A rapier could be main-handed/piercing/deadly
- A dagger could be off-handed/slashing/thrown(10m)
But damage die size would start at 1d4 for everyone, and is modified by class abilities and feats. Fighters might start off with a die increase right at level 1, and they (and other classes) might gain more universal ones or group-specific ones as they advance (each class with their own progression), all with a cap of d12 (or d20 if you want a nice capstone for L20 fighters). A feat could increase your weapon die for all weapons or certain groups of weapons by 1. And having the ambidexterity feat means you have 2 main hands.
- off-handed = +0 dmg but can be used in, well, your off hand
- main-handed = +2 dmg
- 2-handed = +4 dmg (or whatever the maths should work out to),
- slashing/crushing/piercing the damage type
- defensive = +1
- brutal(x) = reroll dmg die if x or lower
- deadly = extra damage die on crit
- etc.
Oh, and a teacup is off-handed/piercing/innocious(advantage on skill checks to pass off the weapon as harmless), and Riddick's weapon damage die with, wel... anything, is d20.
I'm going to add to the "probably too much abstraction" and toss on the addition of how this would massively cut down on the gms ability to make magic weapons/treasures by shifting so much of it into the class where it doesn't really belong. It's bad enough monks already kinda do that, making the other classes do the same would be a mess.