Stagmitis,
With the thumb ring the index finger puts light pressure against the arrow holding it to the riser. If I were to switch sides it would be pushing the arrow off the riser. There are some who shoot off the knuckle side of the rise with the thumb ring I'm told, but I've not seen anyone do it, or any good videos of anyone doing so. I agree that the changed sight picture might be of benefit,better yet would be if my eyes would quit changing dominance, this getting old and getting bi-focals sux. Lajos shoots 3 finger split, and quite well at that.
Bob