Please allow me to quote you: "That being said, I don't understand why China either can't or won't verify degrees the way the rest of the world does with either a transcript or even a phone call."

My response to that is that it would be opening up a real can or worms. If China were to verify degrees, they'd find themselves in quite a dilemma. Fake degrees abound. Wish they would, though. Now it's common practice to work alongside someone with a fake degree, even a Doctorate, who is, even if it's their first teaching gig, making more money than most of the rest of the staff simply because of their falsified credentials. Funny thing is, the Western staff are quite aware of the scam. It just takes a little discernment. Well, no one wants to be a snitch, but sooner or later it leaks to administration. They know and don't care. Looks good to have a few PhDs on hand.

I've been carrying official copies of my transcripts around China for more than five years. No one has ever asked to look at them. I doubt my degrees are carefully scrutinized either. In fact, I have this feeling that at the college where I work they use the dartboard and blindfold method to determine who's going to teach what. There's no rhyme or reason. So, instead of carefully examining CVs and Resumes to determine who would be best suited to teach what, which it's obvious they're not doing, I can only imagine that the Dean has a dartboard with everyone's name on it. Gets out the darts, puts on the blindfold and there you have it. John Smith is going to teach writing this term. Doesn't matter that John can't write his way out of a paper bag. Nor does it matter that John hasn't the least interest in teaching writing and, as it turns out, spends the time in class lecturing to the students about how they should want to be great writers, without actually having them write anything.

Ok, I'm digressing...........

'That being said, I don't understand why China either can't or won't verify degrees the way the rest of the world does with either a transcript or even a phone call.'
Don't use what happens at the provincial education department controlling a developed city like Shenzhen as the model for what happens elsewhere in China.
The problem you have is that you need a foreign experts'certificate in order to get a residence permit.The provincial education department head office staff in Shenzhen is ill-informed re Western degrees/diplomas,so the foreign experts'certificate won't be issued to you by said office at present.
The girl at your school knows about this because she's encountered you kind of difficulty before when applying for a foreign experts'certificate on behalf of a foreigner.And I'd expect it to be very hard to persuade the director of said office to change his view re unsealed transcripts.
But I would expect most other provinces in China to have better-informed provincial education department head office staffs.You can look for teaching vacancies elsewhere in China ,but don't delay,or else you'll limit your options.