What I like: I am authenticated by some proof of ownership, and, unlike with password authentication, the "thing I own" is never transferred over a wire or stored at the server. This is even better than what I have now (unique passwords for every site, managed with 1Password, and 2FA for important sites).
But the SSH callback would have to be automatic - Via a browser plugin, or even better, implemented by the browser itself. I don't think I want to start a terminal and past a string every time I log in to a site.
The "paste this into your terminal" also opens up a completely different can of worms - Social engineering!