Sorry out of context but this may not be about those foreigners you are saying it is. Do you mean that nobody should work illegally? If so then I agree to a point when it comes to those who have their Residence Permits which may not be for work. Those ones have come legally and are legal till found working. They are asked to GO BACK HOME to get their Z visa which means switch their current (not for work) Chinese visa. So, the situation on mainland is a bit more complex than you seem to think it is. Whether schools in China get all the foreign staff they need (legally) or not may be seen soon when the semester begins.

Foreigners who have entered or remain in the country illegally.