With my changes it becomes difficult to actually dump Strength too low due to the requirements on all weapons. In addition, at 19 Strength, players gain an additional die size added to damage over a character that dumps Strength. This means that a 20 Strength character with the Dual Wielder feat is able to wield two Greatswords at 1d10+5 damage each. While a 20 Dexterity character with the Dual Wielder feat is only able to wield two Rapiers dealing 1d8+5 damage each.
One of the issues with the game currently is that you pick either Strength or Dexterity as your main stat while dumping the other, and I don't see why they can't both be as important as each other.
An additional benefit to this is that if you decide to make a Strength and Dexterity focused character, you can wield a Greatbow or a Heavy Crossbow in one hand.
I think these benefits are flavorful for Strength as well as covering a bit of the imbalance between the two.
If there is something I've overlooked, please let me know as this is still the first draft.