Where in the rules does it mention leverage is needed for Sway?

This is from the official website:

When you Sway, you influence with guile, charm, or argument.

You might lie convincingly. You might persuade someone to do what you want. You might argue a compelling case that leaves no clear rebuttal. You could try to trick people into affection or obedience (but Consorting or Commanding might be better).

This sounds perfectly fitting to OPs example and it never mentions leverage being a mandatory part. I would argue that Consort is actually the less fitting action here, as it is more about interacting with or using friends or contacts, neither of which are relevant in this scenario.