These sort of unofficial background checks are unethical IMO, because there's often zero trace of them occurring. That person is essentially blacklisted without any hope of remedy. Unfortunately even where they're illegal they still happen sometimes and the candidates are none the wiser.
I agree with you on workplace safety, but in this case it seems like posturing, since the SARS-2 vaccines don't protect from sickness and transmission (any more). Several countries are experimenting with "vaccinated, cured or tested" to enable reasonably fair and safe access to social life. Since tests will no longer be free, this seems to be a very strong incentive to get vaccinated.
In countries like the UK, Germany, France, etc they're also pondering making it mandatory for health care professionals to get vaccinated. That probably makes sense, together with e.g. teachers, especially since kids can't get the vaccine. But the government must guarantee that any vaccine side-effects will be made up for. The nasty thing about AZ for example was that you would of course get treated if you got the brain thrombosis, but no compensation was awarded. I indirectly know of someone that had this critical side-effect and they survived but last time I heard they were still half paralysed.