Simple: security keys combat phishing, two-factor apps do not. A security key doesn't just authenticate you to a site: it also authenticates the site to you.
evil.com can still claim to be mysite.com and trick users into using their security key there, no?
It would presumably have to trick the user into "registering again" since there is no valid key handle for FIDO2 (not sure about classic U2F), but I doubt that users would notice the difference in flows.
The real win is that this spoof would still not give evil.com a credential scoped to mysite.com, but a user would be none the wiser that they are on a different site than they expect (which can still be problematic depending on the nature of the site).
Even if evil.com gets you to register and present your key, they cannot forward it to mysite.com. Even if they go to all the trouble to completely mitm you and the site looks identical to mysite.com, they cannot get the emails or get it to send money.
This is because evil.com cannot pretend to be you when they interact with mysite.com no matter what you've given them.
It's going to be hard to trick me into thinking I've logged into my gmail if none of my emails are there!
Unless they somehow convince you to put your yubikey in the mail and physically send it to them...
I'd just be careful about overly relying on this property or calling it anything like mutual authentication:
If an attacker can make an educated guess about a user's account contents, they could still convince them to provide additional personal information once they let their guard down after authenticating.
I have never seen a solution like this, but I can't see how it would avoid simple relay phish, it seems as though bad guys can just get themselves a legitimate encrypted barcode and have their victim scan it.
Maybe you haven't explained this solution properly, do you have a link?