the whole procedure, which as you say involves a medical and a Resident Permit, the latter only being issued by the local PSB if the medical report, including an HIV test, is satisfactory, and a Foreign Expert's Certificate is presented along with the medical.

Yo, but at the local PSB, the countersigned employment contract and a photocopy of same must be presented, and of course your passport must have an in-date Z visa, up to 30 days from your date of entry to the mainland.
But if the school decides not to offer you a contract, signed by the school itself, the whole process stalls, and your Z must be replaced by an L inside China before it expires, or else you'll be in big trouble,@ 500 RMB/ overstay day fine plus likely deportation.
Many schools will expect you to start teaching before you've got your RP - illegal, but they're holding all the aces, so you'd be better not to refuse to do so!