THESE HAMPSON PEOPLE ARE LIARS, AND LIE TO ENCOURAGE YOU TO BREAK THE LAW.

Yup, and sadly they're not the only ones. Plenty of school bosses will say anything to entice you to work for their crumby outfits.
But if you don't qualify for a Z visa, try some backwaters in Sichuan, such as Hejiang. You can work there unimpeded on a student visa or an F or an L visa. Usually, rural police don't give a damn. That's a fact.