I haven't thrown in my opinion yet, so here goes. First my credentials -- electrical engineer (takin physics...), and ~45 hrs as private pilot.
Lets establish that the wheels are "free". Unlike your car, there's nothing driving the wheels, or preventing them from rotating. Except brakes, in fact mainly use brakes to taxi around.
As stated earlier, you apply high throttle to start rolling, and then back it off to control your taxi.
If you've taken physics, you'll remember there's 2 kinds of friction. One you use to start motion, the other is used once motion is started. Once you overcome the first one, the 2nd is much less, hence why you throttle up to start taxiing, then back off so you don't take off on the taxiway.
Then think about a plane landing. There are lots of factors here, ground level altitude, heat, humidity, wind. Planes always touch down at different speeds. Mainly due to wind. Compared to the airbody, it'll be approximately the same, in relative to the ground, maybe not. Easiest way I can think to explain is if the wind is blowing strong against the wing, you don't have to be going as fast to get enough airflow over the wings to generate lift. (and you always land in to the wind because of this, every runway has 2 directions to land, and most airports will have multiple runways going different directions to better support this)
So with all that in mind, the wheel is free rolling. And has to be. If it was on any kind of gear or something, it'd flip itself, or destroy the gears as soon as it touched down.
Now. Planes have either a prop or jet, and both of them push air back to make the plane go forward. So they're pushing the body of the plane against the air. Once it overcomes air resistance and wheel's friction, it'll start moving forward.
The treadmill matches the plane going forward, and moves the opposite direction. This force is applied to a free-spinning wheel.
I drew a little picture as I understand it (attached). The end result is the wheel is going to spin really fast. But the treadmill is NOT opposing the jet or prop, therefore plane moves forward, generates lift, etc.
In the non-perfect world we live in. Yes, there's some frictional resistance on the wheel and bearings, etc, but it's minimal, and the prop or jet will still overcome it.
Therefore, I think the plane takes off
