Perhaps you meant to say that the only way to turn a bike in a particular direction involves countersteering. My scenario required no leaning.

If you are on a moving bike and you turn the handlebars to the right you will soon be on the ground. :) On a two wheeled vehicle the handlebars are used to adjust the angle of the bike/rider system with respect to the ground. The handlebars are not used for turning but for adjusting things so that a turn can occur. "Countersteering" is something that occurs at all times and in all situations. It has nothing special to do with turning.