However, the treadmill would cause them to turn in the wrong direction and stop the plane moving.
How, exactly? The wheels are free-turning, so how can they stop the plane moving?
If the treadmill worked do you not think aircraft carrier designers would have fitted them rather than enormous runways?
Airplanes do need enormous runways. Whether those runways are made out of tarmac, water, ice, ball bearings or a giant treadmill is irrelevant. As long as the runway is long enough for the aircraft to reach takeoff speed, it’ll take off.
I think you think we’re talking about vertical takeoff – we’re not.