Keep in mind this is applying your STR/DEX Mod as damage once per turn.
You are granting the same bonus of Two Weapon Fighting Style (Adding ability mod to damage) to Great Weapon, except rather being limited to once per turn, you are applying it to every single attack.
This means as say a hasted 20th level polearm master fighter you could apply it... 11 times in a turn? (4 attacks, 4 action surge, 1 bonus, 1 haste, 1 reaction).
Even with a far less optimized exploitation, you would be applying it at least 2x a turn at level 5.
Also you designed the new Great Weapon Style to work fantastically with the feat for great weapons, but designed the archery style to be worthless with the feat. Fighters have lots of bonus ASIs, they will be getting appropriate feats.
GWF is finicky and weak, but it shouldn't be 2-11x better than the other option.
Archery style is too good at negating the penalty of Sharpshooter feat, but it shouldn't be reduced to non-existence. (most archers would likely take defensive since they would be getting the feat anyways)