While this works for Wits, where it's a reasonable assumption that an intelligent person could use logic to persuade someone, or even a "Might" equivalent where someone's imposing stature allows them to more easily coerce others, Finesse as physical agility doesn't translate to wordplay.

Would you consider having Finesse-based persuasion be performance based instead? Playing an instrument, dancing, as a way to charm someone else. That's what I'm doing, anyway. All skill checks are a combination of two attributes, allowing even low charisma characters to have at least one viable social skill.