I feel like we're missing the elephant in the room.
The solution they have isn't an authentication system. It is a system that relies on an existing authentication system (your email provider's system) to work.
This is essentially the same as OpenID Connect: you really log in with your email host, be it Google, Yahoo, whatever. It is better, because your websites don't have to support the use of a dozen different vendor-specific login buttons.
But the real authentication happens when you enter your password to access your email.
Therefore, when I read:
> I think a better way to solve this is at the browser/OS level with built-in password generation and management.
All I can think is, it depends on who you trust: the entity that secures your mail, your contacts, your calendar information, or the entity that secures your passwords to get there?
We can have a world where identity is federated through your email account. We can also have a world where it is federated through your browser account. They are fundamentally similar, and I have no way to know which of Gmail or Firefox Sync are more secure, but I do know that they're both inconvenient as is.
Gmail keeps nagging me about 2FA, a very bothersome way to log in (since I don't often have my phone), and wants me to have a backup email address for recovery purposes (which I cannot accept from my main identity holder because, well, it is the main one).
Firefox Sync doesn't enforce the use of a master password for all users, which can easily give the appearance of security. Besides, it won't always detect password fields and won't auto-login when it sees them. In addition, it doesn't have the (UI for the) ability to switch users on the browser — you need to use a private window for that, but then your passwords are still shared with other users. Finally, it doesn't generate strong passwords for you (which makes sense, most websites offering a password have stupid rules).
From the website's side, implementing a login solution relying on an existing email provider is way easier and more secure than relying on your browser's Sync system.
So, is it really a better way to solve this? Not currently.