There are obvious latency issues with this though and it would mostly be considered less than optimal today.
Make a national digital ID app. Applicants fill in a web form saying they want to have it. They enter their personal ID number (SSN or equivalent). The one time key for the digital ID is then sent to the mail address. The user installs the digital ID app on their smartphone/computer and enters the key from the mail.
Now they can use that app to 2FA anything: fill in their tax returns online, buy things online, send money to anyone else, login to their internet bank or any other website (by open API), manage social security and other systems, apply for schools etc. This must sound like science fiction in the US but it's been a reality for 5-10 years where I live.
I'm not sure what the fear is with a large central table having a number in one column and your name and address in the others.
That said -if you are afraid of federal/national systems (which is an entirely valid if not entirely rational position) then why have a federal tax authority, national passports etc? It seems like getting the worst of both worlds if you have to run authorities that obviously need to keep track of every living soul in the country whether they like it or not - but then not giving them the tools to do so.
The EU is federal in much the same way (independent states, we don't want the central EU government to meddle too much in state business etc) but then we made sure that the EU government doesn't have anything to do with individuals in the member states, they deal only with the member states themselves. I pay taxes to the EU only indirectly through my home country's tax - so there is no need for an EU tax authority to know who I am, and so on.
The analogous situation for the US would be there was no federal taxes collected from individuals, if states issued passports etc.
If you say "I'm bob smith with SSN 12345", then you gave them your id. You didnt' verify that you are in fact bob smith with SSN 12345.
Now they have to verify with some degree of certainty that you are. So they lookup your address, and send a regular paper mail describing what needs to be authorized, and you sign and return it (or call a number and enter a code, or visit a website, or whatever).
This kind of crude 2FA can still be circumvented by someone stalking you mailbox, but it's a lot better than nothing.
Now: what this requires is a mapping from name+ssn (or some unique identifier) to an up to date address. Having that has several other benefits when it comes to e.g. automatic voter registration etc.