>> Opposition to using the SSN as an identifier started out as a privacy thing. ... They did not want it to become a national ID card.
That's the thing. I understand why that was the prevailing mode of thinking back in 1930s (?) when SSN was first introduced. There were plenty of people without any form of ID back then.
But it's just not so today. Vast majority of people in US have some form of ID, and it is extremely difficult to impossible to do even the most basic things (like, say, getting a job) without some form of ID. Even if there was some valid reasoning behind the desire to not have ID, this ship has sailed decades ago.
The best that can be done now is to acknowledge the actual situation for what it is, and fix the bureaucracy and overhead of dozens of different incompatible IDs. Have one national photo ID, with a guaranteed unique identifier, prominently featuring citizenship / green card status, and with a chip providing crypto for signing purposes and guaranteed secure communication with the government (say, submitting tax returns). Issue and re-issue it for free, using some existing federal government owned infrastructure - USPS seems like the most obvious choice here. Make all federal agencies use this ID for all purposes that require it - taxes, social security, immigration (i.e. replace green cards with it) etc.
States can do whatever they want with their own IDs, but I suspect that most would simply reuse national ID numbers on their driver licenses, concealed carry licenses etc. That would conveniently make all those IDs readily substitutable for most practical purposes.
I also don't see a national ID as a privacy thing in and of itself. It creates a record of you that the government has - sure, but so does your birth certificate (or immigration paperwork for naturalized citizens). The potential violations of privacy begin when government starts demanding ID for activities that didn't require it before, or even just for regular spot checks. But that is a different fight.