The problem with your suggestion is that most recruiters in China (and not only there) are crooks. It's better to contact schools directly and to ask them whether a criminal background check in one's own country is required or not.

You may well be right. However, someone pushed for time and in a spot of bother could use a recruiter as a filter to filter out the places that definitely require the police check. Then he can focus on the possibles, that is those that the recruiter said don't need the police check. I would think there will only be a few, if any. He can apply direct to employers in those areas without informing the recruiter. Just because he has contacted a recruiter doesn't mean he has to accept any job offers through that recruiter. A bit sneaky, maybe. But as you said, recruiters are mostly crooks!