The airplane/treadmill problem seems to assume that the wheels provide thrust for a plane that is trying to take off. They don't. The thrust is provided by the jets (or propellers) against the air, not against the runway. Spinning the wheels of an airplane doesn't result in air flow over the wings. A plane becomes airborne due to the Bernoulli effect of airflow over the wings, not because of spinning wheels, or any other motion with respect to the ground. The arguments that I've seen offered about this so-called problem seem to completely lack understanding of how aerodynamic lift actually works.
Because the jets exert thrust against the air, and not against the runway, the fact that the runway is a treadmill - assuming that could actually work - would be to hold the plane stationary in the air, and there would be no resulting airflow over the wings, and thus no resulting lift. The blowing of the jets (or propellers) would have all the resulting upwards lift of a box fan.
Actually, the more I think about it, I'm not at all sure what assumptions folks are making. But they don't involve any understanding of lift. If the wings don't move relative to the air (or vice versa) there will be no lift.