You can’t remove your last option. It’s probably happening because her Apple ID is her real email so she’d have to first change that which is way unintuitive.
I think it’s best with Apple to log in with an iCloud email address even if you don’t use it.
Oh you're right, you can't remove the last option (just confirmed by trying).
It's the worse case then, and she'd have to change/remove her email from her Apple ID [1].
Apple should do better and make it easier in this kind of case. They already allow one to deregister a phone number from iMessage via a web page, how hard could it be for them to do the same for an email...
I was under the impression it’s [1] possible but I’m away from my Mac to check.
Worse case one could change one’s Apple ID’s email.
Apple should make this stuff more obvious though
[1]: https://news.ycombinator.com/item?id=38780442