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.