That all makes sense, except if someone’s SSN changes (which happens under certain circumstances), doesn’t that invalidate their primary key or require a much more complicated operation of issuing a new record and relinking all the existing relationships?

Yes, in the case of duplicate SSN assignments l you would need to change their records to align with the new SSN while not changing the records that go the the person who keeps the SSN. We do it with state identifiers and it is a gigantic pain in the ass.