That was my thought initially as well, but SSNs are basically a national ID already, no? PKI or blockchain could actually provide citizens with more control over their identification than they have now.
Of course, I don’t expect the government’s implementation to favor the citizen so much… but hypothetically it could be an improvement.
Estonian ID card example https://en.wikipedia.org/wiki/Estonian_ID_card - mandatory document for regular identification, travel document, online identification + digital signature (state and business services built on these). Private key never leaves the chip on card, but as recently discovered vulnerability in chip shows, you have to be prepared for flying brown stuff.
Security is a process.