Earlier quoted context omitted.
Although the typical design for a FIDO key uses what is technically a symmetric ("secret") key that's an implementation detail and the purpose of the FIDO device is to maintain asymmetric (private) keys. If you've been under the impression that FIDO is just a shared secret system like TOTP then I've got great news, it's much cleverer than that. By not relying on secrets the system is robust against total incompetence…
You've totally lost me, and it's not because I misunderstand FIDO. The private key you refer to must be kept secret or the credential is compromised. Yes, this is undoubtedly better than passwords, because the system automatically prevents credential re-use across services and is more resilient than a password hash, but it still requires secrecy. You're not providing an example of a system that functions without the…
Society blurs this line a lot by telling people things are "Private" when in fact they're only a secret, and then there is an opportunity to betray them. This happens for payment cards for example, bank representatives have been known to tell even a court of law that bank employees can't find out your PIN, so if a PIN was used it proves the customer was negligent or actively participated in the transaction. In fact, of course, the PIN is a secret, so the bank and thus its employees are aware of the customer's PIN and an insider could in fact perform transactions using PIN verification despite no negligence by the customer.