Earlier quoted context omitted.
This is the problem with having the public and private key be the same. Anyone should be able to access your public key, and anyone you deal with should be able to ask you to use your private key to verify your identity. The problem is when that entire process is reduced to "give us the number the government uses to ensure you're you. Don't worry, we won't use it to convince anyone else we're you ;) Or leak it so any…
How much would it cost to give everyone a device from which the private key could not be removed? Worried about "mark of the beast" based objections? Make it optional. Those who wish can retire their SSN and receive their public / private keys and then the government publishes their SSN as a trashed SSN. Everyone who still wants just a SSN can take their chances.
USA passport for my children didn't require SSN. And a passport complies with TSA id checks.