I am not sure I follow your reasoning. Key loss would require re-issue - under the present system that uses real details or a biometric these cannot be revoked

I am not sure I follow your reasoning. Key loss would require re-issue - under the present system that uses real details or a biometric these cannot be revoked so cryptography seems better. The current system the UK uses is a Nation Insurance Number, which is just a key in a government database but once issued it is not revokable either. Solutions like a back up reissue key, or quorum of those who signed you key verify it is lost and then it can be resissued, or an trusted official like ones lawyer - or you have to take some ID to a post office. It may be inconvenient but less inconvenient than identity theft today. As for privacy this is solved by crytography too, so I don't follow your first point. If I wanted to message you privately, I encrypt the message with your public key and then you and only you can read it - just like PGP today. Of course this hypothetical system unbackdoored cryptography would be fully legal so we could have keys just for privacy between us / or verification. A bank could issue a key per account, the key ensures your banking cannot be eavesdropped and ensures the identity for this account.