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The trick is to think of an airplane as if it were a car. A car pushes the ground with its wheels; An airplane accelerates by pushing air. That difference decides the whole riddle.
An airplane is on a runway that is, in reality, a conveyor belt. The belt moves rearward and adjusts its speed to match the speed of the plane forward.
The airplane's wheels rotate freely and do not propel the device: they only allow it to roll. The air is still relative to the ground and the runway is long enough.
If the pilot accelerates the engines to takeoff power, will the plane take off?
Answer: yes, the plane takes off. The conveyor belt moves the ground backward, but the plane does not propel itself by pushing the ground with its wheels. Its engines push air backwards and, in reaction, the plane receives a forward force. The wheels just spin freely. If the belt moves backwards, the wheels will spin faster, but that doesn't cancel out the thrust of the motors. As long as the friction of the wheels is small, the belt barely exerts a relevant horizontal force on the plane. What matters to take off is the speed of the plane with respect to the air. Since the air is still relative to the ground, if the engines accelerate the plane forward relative to the ground, they also accelerate it relative to the air. When it reaches takeoff speed, the wings generate enough lift and the plane takes off. The belt only changes the speed of rotation of the wheels. It does not prevent the plane from moving forward. Therefore, under these physical assumptions, the plane does take off.
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