"Clarification", indeed! I've been using the info in your profile at the bottom of each post to identify what hardware you are using! So you also have a newer(?) iMac running
Mojave macOS 10.14.
A. Is that the computer that is covered by AppleCare?
B. So, you have four Macs, including the mini for your wife?
C. You can no longer run Apple Mail on any of them?
I can see my email for a split second each time I open it, does that say it still exists and it's hiding?
Certainly.
Your emails, like files created in other apps, are stored completely separately from the code files of the application.
If you haven't put your home Library folder in the Side Bar of Finder, simply use the "Go Menu" or type shift + command + L. I would suggest using Column View in Finder and just keep clicking the next folder to the right. In your home Library for "Mail", click on "Mail",
then "V7",
then one of the folders with a folders with the 32 letters and numbers "name"
(I have one named: 189FB98D-11AB-4510-8073-B8810130FFAA you are not
likely to have one with that exact name).
You will probably have several of those useless-to-humans named folders. Click on any of them until you find one that has a bunch of items with ".mbox" suffixes.
1. Now click one of those ".mbox" files.
Look for another folder with a horribly long 32 character name. Click on that folder.
Do you now see a "Data" folder? Great! Click on it.
If you don't see a "Data" folder, just click on any other incomprehensibley named
folder until you find one with a "Data" folder in it.
2. Inside the "Data" folder, you will see that Apple decided to save typing by now naming folders with single digits! If you click on one of those, you will usually find yet another group of single digit-named folders. Guess what you will probably find in the
second group of folders? That's right! Another single-digit named folder!
Are you still with me?! For the Grand Prize, what will you see inside one of that
3rd group of folders?
If you guessed another group of single-digit named folders you are to be forgiven. But Apple is not that easy! Instead, you should find a folder called "Messages". Dare you click on the "Messages" folder, you will, at long last, find at least one of your email messages
IF ANYTHING HAS A SUFFIX OF ".emix".
Do a Get Info (command + I) on any .emix file and you will see an email.
Such a short question, such a long and involved "answer".
So, the good news is that your messages are probably OK! All you need to do is get Mail to stay open.