I saw it pointed out in another previous article, but eye motion is largely subconscious. Yes, you move your gaze when you want to click on something, but you also move your gaze when you _don't_ want to click on something, and consciously restricting where you look is very hard.

Maybe if it only moved the cursor and clicking was a separate input it could work. We'd need to come up with different paradigms for interactions that deal with hovering, mouse over/enter/leave, swiping, etc.