No, the site wouldn't know the account, it's the device providing the verification.
I think it's worth remembering that this is a suggestion, and not even originally mine. If you're happier to use the current multitude of age verification services that differ on a per-site basis, with all the security vulnerabilities, risk, and inconvenience that entails, then feel free. Or bypass them using the methods suggested.
I'm literally just providing a better technical solution than has been implemented. What I'm not suggesting is "this is the answer to everyone's problem".
No, I don't know where you've got that idea from.