I wonder if it is a good thing to practice it. In particular, if one learns to sense balance by means other than one's eyes, might that help even when one has eyes open? Sounds plausible, but I have no idea if it is true.
I guess I will try to learn.
But based on the responses, it is clear that I am not alone - many people haven't learned this skill to the extent that the instructor claims one should be able to do. But some of you have.
I wonder if there are correlations between being able to sense and compensate for balance well without vision, and motion sickness. I have had some problems with motion sickness, and they were worst in an aircraft that had no windows, so I could not see the horizon. And I can't spin on the ice very well - could there be a correlation there?
A bit off-topic, but, in retrospect, it is pretty remarkable that cutting only 2 or 3 mm, or so, with a moving skate blade, into the ice, creates sufficient sideways forces to hold one securely and effortlessly upright as we skate. It makes sense in a "displacement hull boat" (like fast sea and slalom kayaks, or most naval ships), that pushes up waves to each side as it moves forward. Each side of the boat pushes against, and is pushed against, the water. If one side lifts out of the water, it no longer has much push, and the opposite side force pushes the boat back upright. (There are also "planing hull boats", like speed boats and many whitewater playboats that mostly push water downwards instead. They handle quite differently.) But why is only 2 or 3 mm or so of ice enough to stabilize a skater? It is stiffer, but maybe part of what the stiffness of the ice does is simply to slow down tilting motions, giving the skater time to compensate...