SSNs aren't a great source of identity mostly because they were never designed to be - they aren't unique, until somewhat recently they were issued in a predictable order, an individual can have their SSN changed, and since they are only 9 digits we're going to run out of them on a practical timeline, etc.
They became a defacto identifier for many institutions just because US citizens and permanent residents generally have one - but aside from convenience, they're pretty problematic compared even to something like a UUID/GUID.
If you were trying to design a true national ID number, you'd probably want to approach the problem more like a database designer would: you'd want something that had possible unique values far exceeding the reasonable number of people you expect to exist before the collapse of society, that was reasonably random and difficult to predict based on other known attributes about the person, that by spec was never shared with another person, etc.