Great point easy, though it might be useful to distinguish between a couple of kinds of stability:
1) A bike's resistance to changing lean angle
2) A bike's tendency to remain in balance in a straight line
In your example of a ski-bike, the front ski would have to have a configuration that mimics trail on a motorcycle. One possibility is the attachment point for the front ski is somewhere in the front of center on the ski, so the ski tracks the way an arrow would. In this case, the ski would tend to re-center itself as you hit small irregularities on the snow. Your motorcycle does the same thing via trail on the front wheel. Inertia does indeed keep the bike going in a straight line and trail keeps the front wheel tracking.
Reportedly (I haven't ridden one), a ski-bike countersteers. It makes sense, as the thing should behave similarly to a motorcycle in that you create a lean angle change by steering the wheels (skis) out from under the bike. Steer the ski to the left and the bike tips to the right.
You'll encounter a little resistance to your countersteering from the trail of the wheel (or ski), but most resistance to a countersteering input will come from the inertia of the gyro (front wheel). The bike has some resistance to lean angle change from the gyroscopic inertia of both wheels, but this won't affect the feel at the bars to the same extent that the front wheel does. In other words, if you have a heavy front wheel and a light rear, the bike will steer heavier than if you have a light front and heavy rear.
To heatmizr's point, the key is in how much energy you put into the bike in a particular moment in time. Let's say you're sitting square in the saddle and decide to hang off to the right. From side to side, your body has no momentum before you begin moving. Then, consider that:
Momentum = Force * Time.
To get your body moving across the bike, you may press with a few pounds' force over a period of a second or two. This won't be enough to change the bike's lean angle noticably, because of the wheels' gyroscopic inertia. Once your body is moving, it has momentum. Now, consider that:
Force = Momentum / Time
If, as you reach the hangoff position to the right, you suddenly pull on the tank with your left knee, you have your body's momentum applied to the tank in a fraction of the time it took to create the momentum. If it took you a second to move across the bike using constant pressure and then you release your momentum into the bike's chassis in a tenth of a second, you will transmit a peak force into the bike ten times greater than the peak force you applied to the bike as you were moving accross it.
In that instant, this peak force may be enough to overcome the wheels' gyroscopic inertia where the force you applied over a longer period to get your body moving was not.
ab