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The key is forward motion, which has everything to do with the tires because that's how the airplane - REGARDLESS OF HOW PROPELLED - moves across the ground.
Yes, the tires reduce the friction between the plane & the ground. But they do NOT propel, aka, cause the airplane's motion. They are only there to reduce friction w/ the ground.

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Let go of how the tires are being propelled (moved forward) and think of it this way: a tire rolling over a fixed surface results in forward motion. A non-VTOL (Vertical Take-Off/Landing) fixed-wing aircraft MUST have forward motion to achieve flight from a standing start.
You are correct. Except that an airplanes tire will not roll unless there is an outside force exerted, because they are not powered tires.

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On a runway, imagine the plane is sitting at point A on a straight line, with points B through Z representing the takeoff roll, Z being rotation (when the nose is lifted for takeoff).

Power is applied and that power - engine thrust - is converted into forward motion that takes the plane through points B to Z because again, a tire on the ground, regardless of HOW it is moved forward, covers a given distance for each revolution (i.e. forward motion). So if NYMadman's observer was standing alongside point A, the plane would leave him/her behind as it accelerated.
Yes. But listen carefully. The tires roll because the plane moves. The tires do not cause the movement. The tires could be fixed in place, with absolutely no rotation, and the plane can still move, aka, a float plane or a skid plane. The rolling tires only reduce the normal friction against the ground. They do not cause the plane to move.

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Distance traveled equals 2πR, where R is the wheel's radius. Let's just say the answer to that equation in our scenario is 12 feet. But the treadmill moves 12 feet in the opposite direction. The tires are attached to the plane, we can all agree on that, right? So if:

2πR = 12 feet but treadmill motion = -12 feet then the resulting forward motion is ZERO.
Wrong. If a treadmill moves 12 feet in REVERSE, the tires that are sitting on top of the treadmill move 12 feet FORWARD. Look at a simple gear setup. Place two gears in contact with each other. Now turn one gear. The other turns in the OPPOSITE direction.

So for your math problem, if the plane moved 12', that caused the wheels to move 12' FORWARD. If the conveyor moved 12' in REVERSE, it moves the wheels 12' FORWARD. The total distance the circumference of the tires have traveled is 24'.

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For the plane to move forward, 2πR MUST be greater than the distance the treadmill moves backward, but the scenario says the belt matches the plane's speed, so the result can't be greater than zero.

On a runway of course, there's no counter-motion backwards so 2πR is always greater than zero, resulting in forward motion.
Your math is wrong. See previous paragraph.

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In the scenario presented, points B through Z are on the treadmill, so as engine thrust is applied, it is NOT converted into forward motion because the treadmill accelerates as the thrust does.
No. The scenario states the treadmill accelerates as the PLANE accelerates. It does not say the treadmill matches the thrust. It says the conveyor matches the plane's speed. Thrust is not speed. Thrust is a force. Speed is a distance per unit of time. They are not the same thing. That's in the given initial conditions of the problem statement.

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Points A-Z simply pass under the plane, and it remains relatively stationary beside the observer, NOT leaving him/her behind.
Your visualiztion of the points means nothing. The plane has to move in order for the conveyor to move. If the plane is moving, then it's moving. There is no stationary plane that's also moving at the same time. There are only 3 dimensions in our physical world; if the plane is moving on a horizontal plane, it's either moving or its not. There is no possible way it can be moving and not moving at the same time...

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No relative wind (airflow over the wings), no lift, no takeoff.
There is airflow over the wings as the plane moves down the runway...

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For the "what if it was on perfect ice" argument, that doesn't fly (ha!) because even though the tires might not rotate, there is still forward motion when thrust is applied because the tires' purpose (reducing friction to allow relatively easy forward motion) has been eliminated by the ice's absence of friction. The plane would still cover points A to Z on the takeoff roll.
How can you possibly realize that the tires are only there to reduce friction to allow relatively easy forward motion in one sentence, yet still hold onto your belief that they propel the plane forward??? There are only two lines of thought here:

1) The engines provide horizontal force to propel the plane. The tires are there to reduce friction between the plane and the ground. (plane takes off)

or

2) The engines provide horizontal force to propel the plane through the air. The tires are there to propel the plane on the ground. (plane doesn't take off)

There is no other viewpoint for the solution. And if you believe #2, that the tires propel the plane on the ground, then you don't have a clue how an airplane works.

Remove the wings. Remove everything except the body, the engine attached to the body that provides the horizontal moving force, and the wheels that reduce friction. When that's the case, it doesn't matter what the ground is doing; whether it's moving or not doesn't matter, because the wheels are only reducing the friction between the ground and the main body of the vehicle.