Is there a way to disable Contacts > Groups button?
Unfortunately, no.  You cannot disable the Groups button on Contacts.
However, there’s a few “operational” things I can think of that might solve the problem.

Why is he spending so much time in “Contacts” to begin with?  Find out what he’s actually trying to accomplish and go from there.  For instance, is it just looking up contact info to make a call/send a message?
If so, then “hide” the Contacts App in a folder.  It can be launched from within those Apps themselves.  This might reduce his temptation to keep going in there.  In fact, the only time I go in there is if I have mass edits to make - I hardly ever use the Contacts App itself anymore.

Create a list of Favorites (Phone App).  This can be used for both calls and messaging; you can have independent  entries for calls and text messages for the same contact.  This will limit the exposure to the Contacts App altogether.  This way, the contacts show up, but you never see the “Groups” button.

The Groups button is actually “far away” from everything else in the Contacts App screen.  Is this a accessibility issue where there’s a physical/medical condition that’s causing inadvertent taps on Groups (yes/no answer - no need for details).  If this is the case, we can look at different input methods. For the screen to change, it more than just Groups.  It’s Group -&gt; Something Else.
What you can do here is take an image (even print it out) of what the Groups Box should look like, point to where “Done” is so he can exit.  Before panicking...just have him compare the image to what he sees on the screen.

Until screen sharing on iOS/iPadOS becomes a thing you can create short video clips (store on OneDrive or DropBox for easy sharing) of how to enter/exit the places he get’s stuck.  I’m sure you’re finding that you’re repeating yourself very often.  If this is the case, a short video of how to get unstuck might work wonders.

You can use video chat to “show” him how to fix his issue.  It’s not limited to FaceTime.  Use Skype if he has a PC.  It’s free and not a data harvester like WhatsApp or similar.  He can easily hold the phone to the webcam (if he doesn’t have one, get one - that might be the best $40-$50USD you spend this year).

Show him how to take a screen shot and kill an app.  If he gets stuck somewhere, he can take a picture of where he is then use messaging or email to mail you a picture of his “location.” At least from there, you can guide him (you can include these instructions with the picture of Groups settings I mentioned earlier.