It's true that U2F/WebAuth doesn't fix e.g. a CSRF vuln or an XSS vuln where someone abuses legitimate credentials like a session cookie, but this does effectively beat phishing, and that's no small feat.
Because the Web Authentication API is built in to the web browser and uses unique credentials for each origin, it's not possible for a user to authorize a login session for a domain that's different from the one they're _actually_ browsing at the time.
If I visit g00gle.com and sign-in using the Web Authentication API, my browser is going to use my credentials for g00gle.com, not for google.com; unlike me, it _can't_ be fooled by similar-looking characters.
> If I visit g00gle.com and sign-in using the Web Authentication API, my browser is going to use my credentials for g00gle.com, not for google.com; unlike me, it _can't_ be fooled by similar-looking characters.
In the age of punycodes this has become particularly important because the human eye cannot visually distinguish between ASCII and punycode lookalikes - many are visually indistinguishable in many fonts.
webauthn helps with phishing in the same way that U2F does: by binding the keys to specific domains/origins. A phishing site hosted at gmailsecurelogin.com can still steal your password, but the security key is not going to produce anything that's usable on mail.google.com as a second factor.
I'm not exactly sure what your threat model is, then. The typical way this gets deployed is with U2F USB devices with non-extractable keys. "Access to all keys" is not exactly technically feasible.
If your concern is "well, there's always the admin/support backdoor", i.e. a compromised admin account or a social engineering attack on support personell could lead to attacker-controlled keys being enrolled, I'm afraid that's not really something you can solve by just throwing new technology at the problem. However, you'll definitely make it harder to even get to the point where admin accounts are compromised by rolling out U2F or webauthn.
Personally, I'm perfectly fine with a technical solution that solves phishing for everything but the most advanced social engineering campaigns.
If an attacker compromises the credential, the credential is useless, yes? The credential is still customized _per origin_, so you'd have to decap the U2F key to extract the secret. This hardware is specifically designed to make that hard to do.
I'll add in to what has already been said by the fact that this should eliminate relay attacks completely as well assuming you have already visited the site before. The site presumably has a copy of your public key so they can send you messages you can only read with your private key and since you have their public key as well, no MITM is possible.
