It's almost impossible to balance on a single ice skate if you're standing still. But give yourself just a little forward motion—it doesn't take very much—and it suddenly becomes easy. You can stand there on one leg and glide effortlessly half way across the rink. Friction will cause you to gradually slow down, and it's only when you've slowed almost to a complete stop that balancing becomes hard again. Why?
This seems suspiciously similar to the question of why it's easier to balance on a moving bicycle. But the standard answer to that involves the angular momentum of the wheels. There's nothing rotating here. In fact, other than the very slight deceleration from friction, moving should make no difference. In both cases, you're standing still in a nearly inertial reference frame.
Does that slight deceleration somehow matter? Or does the interaction between the ice and a moving skate somehow help you to balance?