My Mate is configured to auto-restart running applications.
Previously, the terminal window honored the rows and columns settings.
Now, in 1.28.2, it starts the size of a match box instead.
What is required to have it honor the profile rows by columns setting again?
The latest Ubuntu Mate release is 24.10. According to
it does not feature Mate DE v.1.28 yet.
If you follow upstream MATE Desktop development, then you’ll have noticed that Ubuntu MATE 24.10 doesn’t ship with the recently released MATE Desktop 1.28
IMHO, this virtually guarantees obscure problems for manually installed MATE DE v.1.28. 
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You might need to go to your MATE Terminal preferences and re-select you previously-saved (and named) preferred profile for default on open.
There is only the Default profile there, and it is selected.
Do you suggest that I create another profile, select it, and then select the Default?
That did not work. The terminal still tries to cram itself into the narrow space between the top panel and one of the auto-started apps.
If launched manually, it launches with the correct window size. Only on auto-start, it is wrong.
Sounds like your user Profile is not "active" at the time that the auto-start triggers the MATE Terminal.
My assumption was that auto-start was a system-level, not user-level, function and so, consequently, started using root preferences. If such is the case, you could "become root", change the Default setting for root, then see if that resolves the issue.
Note: I am only a user, not an expert. 
Do you mean running startx as root? Boy, am I reluctant to do that!
No, it is enough to
ugnvs:@pc/home/ugnvs$ sudo su
[sudo] password for ugnvs:
root@pc:/home/ugnvs# mate-terminal
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Did not work.
root's profile has the default 80x24.
The auto-started terminal's window appears to be 36x6. It clearly takes exactly as much space as remains between the top panel and another app that auto-starts too.
This issue is net-new in the latest Mate. It had worked fine for 10+ years until now.
Referencing this thread it's quite possible the OP is using Fedora with the Mate desktop:
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Moments ago, I locked up Mate by resizing the auto-started terminal's window while other apps were auto-starting. After that, Mate stopped responding to the left click while the right click continued to work. Only ctrl-alt-bs helped.