Very cool thought exercise. My impulse reaction though is that this is not a case of a character ability specifically, but a case of the player being wonderfully clever. The player knows something useful, and puts it to use extremely well.
I might ask for a roll from any of...
- Acting
- Carousing
- Current Affairs: Politics
- Diplomacy
- Enthrallment
- Fast-Talk
- Fortune-Telling (it's about reading people and saying something they might believe right?)
- Performance
- Philosophy
- Politics
- Psychology
- Public Speaking (since it's a general skill of speaking well, I think this would need to succeed by a lot)
- Savoir-Faire (perhaps some custom or another would be enough to make a politician bow-out)
- Sex Appeal
- Sociology
- Strategy (if the situation has military components, this might lead you to something that would depose a politician)
- Streetwise
Of course, the skill called for would depend on how the player is trying to play it. And if they can pull off 6 word sentence that makes me as the GM think that the character they're trying to influence would be so significantly affected, then I'd try to find the right skill and ask for a roll (and give them a hefty bonus for the great role-playing).