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Would think everybody ought to get their own private key, with the government only having public keys. Thus if someone hacked a government server they would just get public keys and not be able to impersonate anyone.

I agree that the private keys should be in the citizen's hands, but then you run into many other issues.

Backups - People who don't back up their data will inevitably lose their private key, and then they would have no way to revoke the existing one or authenticate anything that had used it.
Authentication - You're still going to need a signing key the government would use to validate your private key. Does anyone think the russians and chinese wouldn't have that ten minutes after it was created?
Knowledge - what tiny percent of the American public has any understanding of public key cryptography? I'm quite sure it is a vanishingly small number that wouldn't show up as a rounding error of our population.

I strongly believe that public key crypto is a real solution to the issue of authentication, but it will never work until they force us all to accept chips in the hand or forehead. Even then, there will be ways to spoof it.

The SSN is really just a nickname for you. It is not a password that authenticates anything. It is only a 'secret' in the imagination of people who do not understand identity. Until you can make it a combination of something you know (i.e., the number) and something you have (a token of some kind, be it another number that is generated cryptographically or something physical), you're not going to be solving the underlying problem.

Seems to me there is no pratical way to replace the SSN with a private key. It might be good as a password, but people would still need to have a unique user name or id.

Even if we had a chip that had a private key assigned at birth, and the chip malfunctioned, I don't think this would remedy the need for sometimes changing a person's key....it would just make it harder to change unless the chip could get a new key without surgery.

For hackers sooner or later would phish the private key of some wealthy people's chips. They would then have their own chips replaced with one that imitates the wealthy person's chip when they need it to, but reverts back to the one they are supposed to have at other times....which would be easier to do if there were a way to change the chip without surgery...they just need to figure out how the changer works or get their hands on one.