This isn't about finessable weapons, this is about weapons with the 'Finesse' effect. Who exactly were these weapons intended to benefit? Since the effect is only found on weapons that are already finessable, they don't help people with the finesse feat. And people who don't have weapon finesse aren't going to use a lesser damage weapon, especially since the weapon will be weakened by having the finesse effect on it instead of some other more desirable effect.
I think changing the finesse effect to one of the suggestions below would be a huge improvement, and actually make the effect desirable by some players:
1. Remove the effect from light weapons and add it to weapons that normally aren't finessable. Then change the description and the effect itself so that it makes the weapon usable by players with the weapon finesse feat.
2. Change the finesse effect so that it uses the player's dexterity bonus for damage instead of strength. This would be a big benefit for finesse-users who have higher dexterity than strength.
3. Keep the effect as-is, but add a to-hit and/or damage bonus for players who already have the finesse feat.
4. Add an increased attack speed to the weapon. This one would probably be the hardest to implement.
I think #1 #2 or both 1 and 2 would be the best suggestions.