Only if it's run by the state and free for every citizen. Forcing more bureaucracy on email providers will just make everything more centralized under Google and Microsoft.

It should be a mailbox with E2E encryption where the keys are stored on your ID card. Backups stay on secure servers that are legally protected from anyone including the police and only given out when you're getting a new ID at a government service center, encrypted with the cards public key so a hacker in the card issuing system can't steal it.

Every user gets a persistent address used as their identity, and any number of anonymous ones. Locking someone out would be both illegal and inconvenient for the government if all their official business is going through the mailbox.