Motorcycle courses put a lot of emphasis on countersteering because motorcycles go much faster and weight a lot more than a bike so you need to familiarize yourself with the effect from the get go unlike regular bikes where you generally start by going very slowly and the low bike-to-meat weight ratio means that it's relatively easy to steer by leaning with your body as you mention. As soon as you go over, say, 20km/h it's critical to learn how to countersteer properly if you want to keep control.
Try turning on a 200kg motorcycle going at 100km/h by simply leaning on it, you'll be disappointed... See for instance https://youtu.be/8_5Z3jyO2pA?t=2m24s . Later on the video they show that letting go of the handles and leaning with your body also works but only because it actually causes the bike to countersteer "on its own".
Yep. It is even more pronounced on a race track with long sweeping turns, even higher speeds, and nothing else in the world to worry about# (like traffic). The faster that front wheel spins, the more it wants to keep on doing that just as before (gyro).
You will be physically pushing the left hand and pulling your right to set the bike up for a left turn, not just "thinking" it - so as to speak. Of course it is entirely intuitive, until perhaps you stop to think about what actions you are taking, and then realise yeah that's countersteering all right.
#) It's been a few years for me now and I only got a few ride days in after also completing Superbike Cornering School lessons, but I found the art of riding my (then) 600cc GSXR motorcycle on a high speed circuit (Phillip Island) extremely meditative - and exhilarating, especially after learning the function of the knee sliders to assist with controlling lean.
Try turning on a 200kg motorcycle going at 100km/h by simply leaning on it, you'll be disappointed... See for instance https://youtu.be/8_5Z3jyO2pA?t=2m24s . Later on the video they show that letting go of the handles and leaning with your body also works but only because it actually causes the bike to countersteer "on its own".