A lot of people, including myself, discount the gyroscopic affect as how one maintains stability on a bike or motorcycle. It's more in turning the front wheel, ever so slightly and often imperceptibly, to maintain balance. The slower you go, the more you turn the wheel, depending on how smooth you are in maintaining a neutral balance position. If you think about stability that way, it's very simple to understand counter-steering. I have some experience with counter-steering with 40,000 high speed motorcycle miles; mostly on very twisty roads.
Take your bike and pull the wheel to the left. What happens? You fall to the right much more quickly than if you had attempted to lean to the right. The reason is that the front wheel is going left, pulling the whole bike to the left. It's that translation of the front wheel, not some mysterious force countering the gyroscopic affect, that pulls the bike left and puts you and the bike at a steep lean to the right. Makes sense if the bike jumps left, you're going to fall to the right. Now as you are 'thrown" to the rightward lean, snap the wheel to the right the correct the amount to establish the correct line to make the right hand turn. The faster your going and the tighter the radius of the turn, the harder you have to haul on the handle bar (more force) to turn the wheel, but the less amount you have to actually turn the wheel. The extra effort at speed is due to the gyroscopic force that tries to keep the weel in the original plane (inertia) and partly because you need to accelerate the wheel more (force is mass times acceleration) because you have less time to initiate the turn (because you're going faster).
Unfortunately, few motorcyclists know about counter-steering. It's a life-saver technique on the road. At higher speeds like 80 or when riding double, you really have to pull hard on the bars to run twisties. After a day of that, you feel really tired in the arms and shoulders. I mostly forget to do it on an ATB as the speeds are so much lower. Ned Overend mentions it (his book) as one tool in his kit for running turns.
Al