The Plane would only not take off in the treadmill scenario if the Pilot was throttling back the thrust to match the speed of the treadmill (initial reverse accelleration). Hence stationary no air movement, no lift, no takeoff.

If the treadmill's speed is truly infinite then ya the pilot cannot overcome the reverse accelleration and take off, but in reality the takeoff speed will be the same because the plane at full throttle will advance through the air regardless of relative ground speed.