Earlier quoted context omitted.
You do need the SSN to match up with the name and other personal information like age, gender, address, etc. In that way, it's a bit like authenticating with a common username and a password that is publicly available with the username obfuscated (except in the case of data leaks).
Instead of a unique ID like a SSN, we should be using an identity provider to support such use cases. Imagine instead that you would authenticate with https://login.gov (ideally with your credentials and a hardware 2FA device), which would then attest to whatever service you were logging in to that you are you. You can't rotate a social security number with reasonable effort, and we can longer treat it as a secret, b…
Or just imagine https://login.gov/ passively collecting information about all the services you're logging into.
I wouldn't be opposed to common login protocol—preferably a distributed or federated one—where the government and other parties can add their own signatures to attest that a particular identity belongs to a certain real-world person, and you can choose which of those signatures you present to any given service. However, having the login itself go through a government server would be an incredibly bad idea.