> You could have one "master recovery key", the public part loaded on your device (to be registered during account creation) and the private part sitting in your safe.
This is true, it's possible to have just the verification information present without the key itself.
However, a part of almost every credential enrollment process is verification that the credential is usable and controlled by the enrolling party.
For passwords it's simply "please type the password again", for TOTP it's generating the next number, for email/SMS it's entering a code sent to that address, so for adding a public key, it should include some way of verifying that you control the private key associated with that public key. It's hard to do that without having the private key available.
This is true, it's possible to have just the verification information present without the key itself.
However, a part of almost every credential enrollment process is verification that the credential is usable and controlled by the enrolling party.
For passwords it's simply "please type the password again", for TOTP it's generating the next number, for email/SMS it's entering a code sent to that address, so for adding a public key, it should include some way of verifying that you control the private key associated with that public key. It's hard to do that without having the private key available.