Why not FIX how an SSN can be used? It was not created for this purpose but that doesn't mean how we use it can't be fixed.

Any time someone attempts to use your SSN to identify themselves as you, you should be notified and your authorization should be required for that use to be allowed.

And the higher the value of the authorization, the more care should be required.

Companies are able to do this already with 2 factor authentication. And I think we SORT of have that for change of address, as the Post Office both requires ID and notifies you by mail. Maybe that's really just one factor.

Allow people to require as many additional factors as they want. Confirming my identity in 6 ways when I decide to sell my house sounds good. 4 ways when I buy a car. One or none is fine when I make a $10 purchase. Let me decide.

And let them CHOOSE what organization manages authentication. A private company might do the job a lot more effectively than the Post Office.

Add human methods too. A call from a sibling or child might be one people could set up. Require validation by a notary public.