This means you never send your private key to the site, so if their database is compromised they only get a public key, which can’t do anything (it is public, after all)
No amount of phishing can get you or your client to hand over your private key, because there is no use case where that is required. There is no possibility of MITM, since your private key is never sent anywhere in any situation.
As for getting locked out of your account, losing your passkey would be the same as losing a password. You would need to follow the site’s password reset flow, and if you can recover from a lost password, you can recover from a lost passkey.