It's my understanding that it's illegal to work directly for a school, that's how it was
explained to me by the helpful police officer: as a FT (on both the study and the
working visa) you must always operate through a third party, and they must pass it
through the local government operator, E.g. FT -> Contractor -> Local Gov't ->
School.

Your understanding is totally wrong. Enfly have led you straight up the garden path.

Why do you think direct there are direct employers in China?

I for one work for a school where we only employ directly. No third parties, agencies or recruiters are involved at all. We do everything ourselves from recruitment, HR and administration and it works for us considerably better than when we did deal with agencies. We would never go back down that road.
School.