Correct! A rubbish school is just that. Makes no difference if the teachers are "native speakers" or not. Private schools in China are crap, that's all there is too it!

Half of us work for private outfits, even if our place of work is a state school.

Just because a school is rubbish as a school ,it doesn't mean that it's a bad place for an FT to work. Many rubbish schools give you a good flat-pay for your aeroplane-pay for everything and treat you really well.