FWIW I'd like to see a couple changes-
Str-
Damage bonus to weapon should vary by STR relative to weapon weight. You only need so much strength to use a dagger as effectively as you possibly can (limited by arm speed, etc)
Daggers get bonus from dex actually. And they're the only weapon type atm that you can use to backstab.
Weapon type should have an effect on damage bonus.
They do. Different weapon types = different requirements, different damage ranges, smaller/bigger ranges, different chances to hit crit, different crit damage calculations.
I'm hoping that a lot more than daggers get a dex bonus. Fencing weapons for sure (rapier,estoc, etc) Basically if its an edged weapon dex should play a part. Weighted by mass. So a small punch dagger might get the biggest dex bonus, while a 2h flamberge greatsword would only see a very small one.
On the damage bonus part, I'm thinking mainly in relationship to the stats, not a innate bonus. IE an arming sword might take 8 strength to wield. do 1d8 slashing damage. Crit on 19-20 x2. That's innate stats. That's typical.
I'm thinking that you might say the arming sword is 8str to wield and gains a strength bonus of +1 per 2 points up to strength 16. It also get a dex bonus of +1 per 3 points up to dex 14. Its rate of attack could also be modified by this. Thats a fairly average weapon however. Say you have a rapier. It might be str 6-12 +1 per 3 Dex 8-16 +1 per 2
It would allow a much large range of weapons to be effective depending on how you balance your stats, rather than the normal min/max thing you need to dump all your points into one stat to be effective.