I open a thread asking about this a while back. Unfortunately disabling options as described above from the Software menu didn't worked out for me as I wanted to, but I rarely see any notifications and was okay with it. If you look into this answer it's suggested something a little more advanced but I never actually tried it:
Note, however, this is for Zorin 15.3 which is the previous version. I do not know if this would work so maybe wait for confirmation from someone?
I've installed Zorin in a few machines in my family and had the same concern. I instructed them to always install any updates that come up at startup, and so far had zero issues with it for a couple of years now. Since you want to take control personally over it you can simply instruct them to ignore it and close it always.