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.

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.

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.

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.

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.

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. Points A-Z simply pass under the plane, and it remains relatively stationary beside the observer, NOT leaving him/her behind.

No relative wind (airflow over the wings), no lift, no takeoff.

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.

BTW, asked my wife so as to get an independent opinion, and hers was profound: "Don't know, don't care. You guys need to get a life." Yes, dear.