There's no reason not to keep SSN as an identifier. Just the same as I wouldn't change my name if I suffered identity theft. Instead there needs to be authentication (eg via a method such as a token, 2FA or whatever) - and it's that which needs to be resettable.
Simply making use of SSN alone illegal in certain industries would be a reasonable approach: it would stop current problems whilst not insensing the "mark of the beast" brigade.