First, exit quarantine requires a health certificate (depending on your visa), and some do not. This health certificate should be handled at the inspection and quarantine office at the exit port.
Our one-year work visa to Russia requires a health certificate after quarantine at Suifenhe China Customs, which is handled by the local inspection and quarantine bureau, but Russia does not want it.
Second, in the destination country, the local immigration bureau or epidemic prevention department will also need local health certificates to apply for work permits, that is, work permits and the like, and will also check them in the future.
We have a one-year work visa to Russia, and we will apply for a Russian health certificate before the visa comes down.
Your local health certificate doesn't work. It is not run by a hospital. It's for inspection and quarantine. To leave the country.