Interestingly, I could restore the mail app functionality by clearing the app storage and cache (as mentioned in this thread) and adding the mail accounts back manually, i.e., without a using a mail settings backup. (I did not lose any mails in the process because they are kept on the mail servers in my configuration.)
Still, this is just a workaround for a problem that I hope will be fixed by the /e/OS developers. In fact, another user has already opened a bug report here (issue #7624).
Note that Advanced Privacy - tracker protection is active on my device, but hide my ip is inactive (probably similar to @irrlicht’s configuration).
EDIT: The problem has reapparead after a couple of days, reason unknown! See my comment below.