You can be annoyed all you want, but the process is insane. The only options available to me are my two university emails. One of the email domains simply no longer exists. I actually took the time today to call my old university, and they were kind enough to temporarily turn on my old university email address for the other email tied to my Facebook account. Facebook won't send the code to the email after I correctly login. It reports an error. (I verified that the email is working.) There's no way to report the error anywhere. The only way you can report an error to Facebook is to be logged into Facebook. There's another login screen that allows you to reset your password. I did so there since that actually worked by sending an email code. After I do that, it takes me back to the "browser verification" screen. Now, the email I verified has disappeared from that list! The only one left is the email that is tied to a domain that no longer exists. It is impossible to get an email at that address. It's ridiculous.

TLDR: I actually recovered my old university email address, and Facebook refuses to send a code to that email and has now removed it as an option.