How would this work? What happens if a child picks up my unlocked phone and copies the authentication data to another device?

I guess you can put the proof-generating code inside some kind of a secure enclave? But then it's still not any better than classic asymmetric exchange, except that the government provides you a certificate that signs the private key held inside the TPM.

Or are you thinking about using a ZKP for a biometric proof? But then this still doesn't solve the issue of a malicious user just taking biometric pictures once, and then re-feeding them to the verifier.

I don't think this is solveable without some kind of trusted computing environment, at which point the classic asymmetrical crypto is fine anyway.