The mechanics are the medium through which players relinquish control of their roleplay. They're sort of what mediates roleplaying, I think, by providing a sort of arbitrary and neutral interface to both limit and move along characters.

Precisely. Which is why it feels different from improv. Rules and predictable-but-random outcomes add verisimilitude to a game.

And the closer something is to the game's core experience, the more likely it is that a rule (instead of a ruling by the gm) is needed. So if you're playing a game that revolves around intrigue and seduction, you will probably want some rules for that. They don't have to be overly complicated, but they will need to exist.