No. That didn't work.
However... I did a little hack. I made a temporary change to /usr/lib/python3/dist-packages/aptsources/distro.py and changed template.distribution == 'Ubuntu' on line 91. This fixed the problem sufficiently to bring up the Software & Updates dialog. In fact, in the enumeration of distribution_id, none of them specified "Zorin" which is why the original exception was being thrown. I think I need to debug backward and see where the sourceslist is coming from.
Incidentally, although the Software Updater still says "Settings & Pro", clicking that does actually launch software-properties-gtk. I don't know why. My guess is that there's a Zorin specific version of update-manager but I'm running the base Ubuntu version. Does that make sense? As I said earlier, I had to force 1:22.04.22 rather than 1:22.04.21+zorin1 on it. Possibly that was the reason?
I believe the ZorinOS version button just says "Settings" only.
Screenshot illustration from another thread here: Error when running Zorin 17 Software updates
Yeah. It's because I'm using the wrong version of update-manager.
I fixed that by sudo apt remove update-manager update-manager-core python3-update-manager and then sudo apt install update-manager again. This installed the correct version with the "Settings..." button.
It doesn't fix the original issue that lead me to hack distro.py. I still need to debug this sometime today. Good thing I'm on holiday this week. 
Keeps you away from the mince pies 
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Problem Solved!
The issue was the wrong version of python3-apt and its distribution list, specifically the files under /usr/share/python-apt/template. These were missing several info/mirror files including Zorin.info, which was due to the fact that I had the stock Ubuntu version installed rather than the Zorin version.
The fix was to apt remove python3-apt and install it again, followed by zorin-os-desktop to restore the dependencies. Once this was done, everything started working and I could remove my hacks to distro.py. (Possibly removing and reinstalling zorin-os-desktop alone might have been sufficient.) I've tested everything, including the Software Upgrade dialog that triggered this thread, and all is working as expected.
Thanks to @Aravisian, @Ponce-De-Leon, @Forpli and others for your patience and guidance. It has been a learning experience!
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