This appears to be a bug.

After adding the email address and clicking save, logging out, and logging in, I found the email address was successfully added. At this point I could change it to the primary one, click save, paste my password, and click the button on the password prompt to successfully change my primary email address. If I tried to add the email and make it primary in one session, it would not work. If I entered my password and hit enter, it would not work.