If "having to" leave the country amounts to being deported, I'm fairly certain that it goes to same category with the Chinese government paying the flight tickets in order to get the foreigner out of here. That is, they would pay to get you shot.
Under milder circumstances for the exit I don't know, but I suspect that having to leave the country will not allow one to get a jab any faster than foreigners here in general - unless one is in position of some added privileges.
For our company registration and subsequent administrative functions, we've had to prepare power of attorney documents on number of occasions, but I don't recall it ever requiring notarization or other official thing - and I believe it is notarization that you are looking for.
Self-written, signed and stamped A4, and our bank and other offices have been fine with it.
But perhaps your purpose for this is different, and it will require notarization - or even legalization elsewhere, if it is for international purpose.
For some personal documents that have required notarization, the office we have used is this:
At least my (foreign) insurance provider provides a "second opinion" service, which might be useful in some cases that don't necessarily require seeing in person.
In any case, if one has such insurance, might be worth to check what is their policy about vaccinating under special circumstances (such as heart conditions), or with the Chinese jab overall. In case there are complications, and you hope to be covered by the insurance.
In some countries the education for police officers is not much more than education of chengguan in China.
It is also that elsewhere it is the responsibility of said police to do what is outsourced to chengguan in China, and for comparative figures one could get more realistic picture by seeking sources with that keyword.