Is that what you’re trying to say?
Granted it still wouldn’t prove it, because we are not our ssh keys. We’re all potentially one malware infection away from having our private keys compromised. Also, if someone wanted to "shed" the identity associated with a public key they could always just "accidentally" leak the private key in a public git commit.
That would allow anyone to prove that they owned the public key, which prevents the original owner from using it. But it seems like, if you want to stop using the key, it's simpler to just stop using it. What does leaking the private key accomplish that deleting the private key doesn't also accomplish?
Huh? I presented the claim to identity that submitting a public key implicitly makes as being the only thing that our hypothetical applicant is seeking to avoid. I valued the technical deniability at zero.
But I said above, and say again here, that most job applicants are not seeking to avoid disclosing their identity as they apply for a job. They are usually specifically trying to highlight it.
I don't believe this is true, right? You do a private key operation demonstrating you possess the private key associated with the public key.
Or, by contradiction: Since the key is public, any server can put the fingerprint of the key in an authorized_keys file. It can then challenge you to log in in a way that exactly matches what a real server you'd actually want to log into would do, because a real server doesn't have your private key either. If your client could also authenticate to the server in a way that didn't prove anything beyond possession of the public key, then it could do the same to some actual server, i.e., the SSH protocol would have no meaningful authentication at all. Because we know the SSH protocol is not completely and trivially broken, this cannot be true.
(I think you also overestimate the value of technical deniability - certainly outside a court of law, nobody is obligated to think, "Well, it could be a complete coincidence, so I'm going to disregard this piece of information I just learned." And I wouldn't bet on it inside a court of law either.)