In that case, third parties could use a government website to get a row/col and ask you to verify, and the website could say yes/no. Yes, there is a risk of your one-time pad being stolen, but it is no greater than the current risk that any US citizen's tax documents or SS card can be stolen.
Who issues the private key? "get" implies it comes from somewhere, i.e. a CA system.
If the government is the CA system, and your private key is your identity, how do you establish your identity in the event that you lost your key?
The nice thing about SSNs being immutable is that none of these are concerns. (It's also the bad thing about SSNs being immutable.)
We do have one thing in the US that’s physical proof, and that’s your birth certificate. But I’m sure people lose them and they can be pretty easily fabricated.