The school does get you a Z-visa

As a matter of fact they are very insisting in you binding your Z visa to their company, they won't accept spouse visa holders or any other type if visa.

I see this not as a proof of honesty but as a way for them to ensure that they have the power to screw you over if you ever decide to leave. If you are under any other type of visa they are powerless to do anything once you left, but if you have a Z visa they can cancel it with a simple call to the PSB, ensuring that you can't easily find another employer in China, hell they can even blacklist you.

I know that it's illegal to work on other visas, but in the case of CGS I seriously doubt that legality is what they are concerned about when giving you a Z visa.