Insurance information changes over time though so it's simpler to just request the government issued GUID and use that to make sure you're associating the patient files correctly.
I think the only real solution would be for smart cards to come back into vogue and assign everyone a unique key along with their SSN. Proofs of identity for things that matter ie loans and other financial work could then be secured without significantly interfering with it's use as a unique identifier in cases where it doesn't need to be secure. (As wrong as it is to use SSNs that way it's ingrained enough in interactions and systems that trying to cut that off would be a big pain.)