I think it's more about keeping the body in balance to avoid swaying to the right side - just keeping the eyes on the ball won't help a lot. When you keep your head down and the eyes on the ball in the downswing you might get problems moving your weight towards the target.
I made exactly the same experience. And really weird is (I saw this a few times) when some people (no Pros) teach beginners to keep their head down... Awful!