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.
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.