SSNs are much too short, and were mostly issued in a foolish and predictable way (if you're a kid you might have a random SSN but most Americans still have ones issued the old way). Given the US plausible population load, issuing a randomly chosen 12 digit number incorporating a check digit would have been a better start.

But the authentication problem is the tricky part though, governments don't have a reliable way to authenticate their citizens today and even if you have a good will intent to fix this, that can be hijacked by people with less saintly motives. See India or China for how that might not go well.