> What is the state of Referer in modern browsers?
Referers are still sent in all popular browers and expose information that is not necessary at all.
> Why keep it?
I can't think of any reason. Can you?
> Would removing it break anything?
It might certainly break some websites. But mechanisms which rely on the HTTP referer are usually broken, anyway. It should not affect too many sites.
> File bugs against browser developers to remove Referer
The referer is such a fundamental feature (not due to its importance but history/habit) that a filed bug will probably be closed within minutes ;)
> Meanwhile users can disable Referer
Certainly. I have. But as with all security-related things, how many "normal" users will do this? 99% of users don't even know about "about:config" in Firefox etc.
> This becomes less of an issue as sites move to https, where Referer is not sent
This is wrong. Even on HTTPS, the referer is always sent unless the referenced site is plain (not secure) HTTP.