IMO:
1. This should only apply to staff who are in regular/prolonged contact with patients.
2. Alternatives such as were used before vaccines became available should remain an option.
What you said is truth and common sense.
What this is actually is: A marketing scheme.
Two immediate family members work at a hospice (not one that is a residential facility). One is in frequent contact with nurses, aides, and various visitors (e.g. for counseling). IMO, vaccination would be a necessity for her, and she is (and would have if she didn't work at the hospice). The other seldom is around nurses, etc., and IMO, vaccination would not be necessary for her job (though she is, by choice).
A facility like theirs or a hospital or a SNF/ALF have people who need to be vaccinated. I also think a business that does not - e.g. a bar - should have the right to require employee vaccination, as silly as that might be, as part of at-will employment. And bartender, server, etc., etc., etc. skills being fungible and in demand, their employees can choose to go with the requirement or move on to one of their competitors.