You're right, the weapon does do most of the work.
Greatsword:2d6+0 for average strength.
Greatsword:2d6+3 for above average
Greatsword:2d6+6 for max (18) strength
The weapon, doing an average of 7 damage...
Accounts for all, 70%, or 53% of the weapon.
With a longsword...
d8
d8+2
d8+4
And now it's all, 75%, or 50% of the total.