I might write up my own version of the eye tagging etc because you are right - there are lots of people asking the same questions (and I sometimes find it hard to find the old discussions). In your case I think the problem is the “Pupil Range” tags are on the wrong layers. If you look at the blue boxes, the “Left Pupil Range” and “Left Pupil Size” boxes are different sizes (there are two outlines), so the left eye can move. The right eye however is not the same - they seem to be tagged on the same objects. As a result, the right eye is not allowed to move.
This is normally a matter of the auto tagger getting things wrong. You need to go through each layer one by one and make sure the right things are tagged. I may have a go at writing up more detailed instructions tonight - if I get it done I will try and post here as well. But hopefully the above helps for now.
I'm not sure what the bug was, but it was solved by relinking the Ai file.
Sorry! Did I not include link to blog post here? D’oh! https://extra-ordinary.tv/2018/04/21/debugging-character-animator-eyess/
My guess there was an extra stray tag in there by accident, you cleaned up.