I have looked at:
I do not use any distro for either Firefox or Thunderbird, but download these from the production release from the developer site (Mozilla, etc.). I then install these as root in /usr/lib . To update, I have configured Ubuntu to allow me to switch user to root with a MATE desktop. As root, I run Firefox or Thunderbird and use the internal mechanism of each to get updates-in-place from the developer, not from a distro.
As such, neither show up as the "default" clients for Ubuntu. Within Firefox, there is a mechanism to make Firefox the "default" browser and that seems to work. There is none that I could find for Thunderbird as the default email client (to IMAP servers inbound, to SMTP servers outbound). The URL at the top of this message shows mechanisms to find files that seem to setup this default; however, it much easier (and safer?) to find a working utility that does what I need.
From the MATE welcome screen, I installed Brave and Vivaldi, not to make these any form of default, but to have these available. Brave seems to have automagically made itself the default until I instructed Firefox to be default.
Is there a utility that allows one to add an option for Thunderbird to MATE Control Center -> Preferred Applications -> Internet -> Mail Reader pull down menu? Is there a file or set of files that must manually be edited?
If I must install Thunderbird from Ubuntu as an application, I will do so, but that makes much more work for me. Assuming the default location from the Ubuntu install will be /usr/lib/thunderbird, I would need first as root to cp -pr /usr/lib/thunderbird /usr/lib/thunderbird-save, then install the Ubuntu distro thunderbird, then mv /usr/lib/thunderbird /usr/lib/thunderbird-distro, then cp -pr /usr/lib/thunderbird-save /usr/lib/thunderbird , and I would have a Thunderbird client that is from the developer, not a repackaging/building by a distro.