Ok, I read maybe the first 20 or so pages, then decided there is too much confusion. Here is my attempt at explaining:
1) Everyone seems to agree/understand that if there is no air movement over the wings, then no lift. This is correct. What people are confused about is whether there is actually going to be any relative movement between the airplane (wings) and the air. The answer is yes.
A simple FBD should be all that's required to establish whether the plane is moving forward or not. I'm not drawing gravity and lift since they have no effect on forward movement, and also not drawing drag, because we all agree that the thrust is going to overcome drag. Finally, and most importantly since
the wheels/threadmill are frictionless they do not exert a force in anyway, so all that was a red herring.
This FBD simply establishes whether the plane moves forward or not. Clearly since there is only one force, then it has to. (Side note: What if there was friction? Well, if the thrust is going to exeed friction, it still moves forward.)
Ok, so what about the whole "wheel moving at the same speed" thing? Well this is a bit tricky. Clearly as the wheel rotates, velocity of different points on the wheel with respect to the runway is going to be different.
Suppose point A, B and C are actually all moving at the same velocity along the runway. Then the wheel is not spinning! It would instead be sliding (or slipping (!) but more on this later) along the runway.
In order for the wheel to spin, in one instant in time, point C has to move faster than point B, which has to move faster than point A! Now what are these speeds?
Point B, which is where the plane is attached to the wheel has to move at the same speed as the airplane, lets call it V1.
What about the speed at point A? This is where the "no slip condition" comes in. It simply means that the wheel is not slipping with respect to the runway. Imagine the wheel "rotating" about point A as it is moving down the runway. At that moment in time, point A has a velocity of exactly 0 with respect to the runway! (ie. it is not slipping on it) Ta da! Velocity of the wheel = velocity of runway. In fact, any point of the wheel that is in contact with the runway at any given time has velocity of zero w.r.t. the runway, regardless of how fast the rest of the wheel is moving.
Hence, plane takes off, wheels are moving at the same speed as the runway, QED. Oh and I got to show of my MS paint skillz.
Edit: However, I do suck at image hosting.
Edit 2: Aha, got it.