I appear to have found the depressing answer to my own question. I'll post it as a cautionary tale that may save someone else from a similar fate.
In Debian, I was backing up my home directory, which contains the Thunderbird profile, using Ark to save it as a tarball. To ensure none of the files were in use, I booted into another OS and did it from there. The process produced nice little tarballs that on cursory examination, looked complete.
It was only by digging into the profile structure and how the data is stored that I was able to identify the problem. None of the files that contained my actual Thunderbird data (emails or calendar), were included. Every backup I made was missing those few files and apparently (hopefully), nothing else. I haven't figured out why those were treated differently.
Bottom line:
- Don't do backup by using Ark from another OS.
- Verify that all of your critical files are included in your backup result.
Speculation: UID might have been the culprit
Both OSes have since been replaced so there is no way to verify this, but it is something to check if you are in a similar situation. I reinstalled newer versions of Mint and Debian. This time, both were on partitions of the same drive, so the first installation was visible to the second, which may have affected the issue. In the process of setting up Thunderbird so that the two could share the same profile (as in @wendy.krieger's suggestion), I discovered that, at least in the new setup, Mint and Debian used different default user IDs (UID).
I'm speculating that in the previous setup, the actual user data was associated with the UID, perhaps with different permissions than the rest of the profile, so it couldn't be seen from the other OS. In the new setup, I had to change the UID on one installation for it to see the profile on the other.