It seems like with this problem, you are just spinning your wheels.
If you put a car on the magic treadmill, it would never get moving at all, because it's the car's drive wheels that push the car down the road. If the treadmill matches the speed of the wheels, the car doesn't move.
A plane is not pushed down the runway by the wheels. The wheels turn freely, and the plane is pushed by the thrust of the jet engines. It doesn't matter what the wheels are doing, the plane will still move forward under the thrust of the engines and will take off.
I'm actually having trouble visualizing how the magic treadmill would work with the plane. What does it mean to say the treadmill exactly matches the speed of the wheels, moving in the opposite direction? As long as the wheels are not sliding against the treadmill, doesn't the treadmill always match the speed of the wheels? The problem doesn't really define a frame of reference for measuring the speed of the wheels.
With the car, it's seems obvious, the car's motor drives the wheels at a certain rate, and the treadmill will turn at that rate. As soon as the car applies torque to the wheels, the car's momentum keeps the car in place, and the treadmill begins to spin at the rate of the wheels, leaving the car stationary.
The plane doesn't move by applying torque to the wheels. So when the plane fires up its engines, right before it starts to move, the wheels' rotation is zero, and the wheel has an angular momentum that will resist it beginning to turn. So when the plane starts to move, could the treadmill just move with it, matching the wheels' angular speed of zero?
Or or does the problem mean that the treadmill matches the movement of the hub of the wheel, but in the opposite direction, like in the earlier free body diagram? The diagram shows how that would work very well.
Or is there a way of looking at it more like the car problem, and before the plane can even move the tiniest fraction of an inch, the wheels and treadmill spin up "to infinity, and beyond!"