> Privacy Advocates would lose their minds
Privacy of authentication may be a valid concern (e.g. during voting), but I don't see how it applies here. If what I want is to confirm to the bank that I am who I am, with all the details about me that I have told the bank already anyway, I very clearly and openly forfeit my privacy. I explicitly ask to be precisely identified.
For banks an other cases that (1) need to know you true identity, and (2) provide no expectation of privacy regarding sharing the existence of accounts with the government, a government run authentication would be fine from a privacy point of view.
The issue is that every site has moved to using 2FA, and most of them have no legitimate need to know your true identity. So using a government ID based solution would unnecessarily conflate authentication and identification and would be a real privacy concern.