Email not sending is a good problem - it means someting is broken and you gotta look into it.
Good thing about email is that it can be dismissed. It literally goes into an inbox where you are already looking at other things that need your attention - and then you can delete/archive when you see that everything is fine.
You can also do more complicated things on the receiving side of the email, with something like imapfilter (or just a thunderbird incoming mail rule on a desktop PC at your house), to auto sort alerts or notices of nominal condition into certain folders/subfolders.
Good thing about email is that it can be dismissed. It literally goes into an inbox where you are already looking at other things that need your attention - and then you can delete/archive when you see that everything is fine.