Are hardware security keys (e.g ones supporting Fido2) “able to protect authentication” even in case of compromised devices?

Correct me if I am wrong, please.

I understand that 2FA (MFA) increases account security in case an attacker obtains a password which might be possible via various ways, e.g. phishing, database breach, brute-force, etc..

However, if the 2FA device is compromised (full system control) which can also be the very same device then 2FA is broken. It’s not as likely as opposed to only using a password but conceptually this is true.

Do hardware security keys protect against compromised devices? I read that the private key cannot be extracted from those devices. I think about protecting my ssh logins with a FIDO2 key. Taking ssh as an example, I would imagine that on a compromised device the ssh handshake and key exchange can be intercepted and the Fido2 key can be used for malicious things.

Additionally: Fido2 protects against phishing by storing the website it is setup to authenticate with. Does FIDO2 and openssh also additionally implement host key verification or doesn’t it matter because FIDO2 with openssh is already asymmetric encryption and thus not vulnerable to MitM attacks?