It would just be another case like the EU cookies thing. Every website would have the button and everyone would click it immediately to get rid of it. It would just be an annoyance.
If done well, the chrome would be more clever than that. There should be "log in as [username]" and "stay anonymous". Unless websites want to start validating email addresses to let you read their content, they'll have to accept "stay anonymous" because it would be indistinguishable on the server's end from getting a brand-new user.
So you'd have an idiotic banner pissing off your users and the considerable majority would click "stay anonymous", gaining the site operator nothing.
So you'd have an idiotic banner pissing off your users and the considerable majority would click "stay anonymous", gaining the site operator nothing.