To be fair: Background checks, in the US, aren't limited to criminal records.
First and foremost, sometimes checks include arrests (without conviction, or a not guilty verdict) if the company chooses the wrong background check company. The government doesn't regulate them very well.
But it can be things like a google search for you (Which means that a protest you went to could mean you "aren't a good fit for the company.").
It can be a credit check: I had to pass a credit check to work at a gas station once. The district manager was convinced that bad credit meant you were more likely to steal.
Very, very often - and more often than a criminal background check - is a drug test. I didn't take a job that did pre-employment hair follicle tests, mostly because the person in the room with me explained she had a drug problem in the past. She had been clean for 2-3 years clean and was willing to take regular drug tests. But they said the tests went back further than that, and no job if the hair follicle test wasn't passed (I think there are now laws that would have protected her).
Urine tests are more common, though. And a slew of employers do random drug tests - some call centers, for example, might do random hair follicle or urine tests (Student loan collections do this in some places). It doesn't matter if it would affect your job or not.
This is all mostly because there is no law protecting it: The behavior would be illegal where I live now.
Now, with vaccinations: Employers do have a little bit of business knowing health concerns if it can affect workplace safety in ways that can make other employees or customers fall sick. They already do this to some degree in, say, foodservice. For example, I know a food factory here (That I worked at temporarily) that requires stool sample tests and time off work if you have any sort of vomiting or diarrhea - you know, because it can make others sick.