This is long, but read it and you'll come to understand why the plane does take off.

we agree that a car (substituted for the airplane for the airplane in the original scenario) on the treadmill will remain staionary to a fixed point on solid ground.

You claim and believe that the plane remains stationary to a fixed point on solid ground when the plane is on the conveyor, much like the car on a treadmill does. However, that assertion is wrong.

The scenario does not say that the plane remains in one spot relative to the ground, only that the conveyor and the airplane have the same speeds in the opposite directions.

The scenario does not say that the plane causes the conveyor to move, but that the conveyor is moved by some other control to match speed of the plane.

So if the plane is taken off of the conveyor and placed onto hard ground and then beings moving towards the conveyor at 10mph, then the conveyor is still going to move at 10mph in the opposite direction of the planes motion...

Now replace the Airplane with a car (make it a Front wheel driven car) and with the same setup as above, but this time the car drives onto the conveyor.
Once the front wheels of this car touch the conveyor the car stops it's forward motion with respect to the ground - with the front wheels on the conveyor spinning at 10 mph on the conveyor and the rear wheels (which were only spinning because the front wheels were moving along the ground and the rear wheels have to follow) are on the ground stationary. We know this is true because the the length of the car and the distance between the wheels remains a constant.

Now replace the Airplane with another car (make it a 2wd Xterra) and with the same setup as above, what happens when the Xterra drives onto the conveyor? When the front wheels contact the conveyor they (being free spinng) double their speed as they continue to move up the conveyor until the rear wheels get onto the conveyor. At that point, all 4 sets of wheels are spinning at 10 mph and the Xterra stays in one spot with respect to the ground. We know this is true because the the length of the Xterra and the distance between the wheels remains a constant.

Now, an Airplane has front wheels and rear wheels, so what happens when the airplane moves onto the conveyor? The front wheels of the plane, which are free spinning, begin to rotate at twice the speed of the rear wheels, which are alos free spinning and on solid ground, and the plane continues to move with respect to the ground...
Does the plane stop when all of the wheel are on the conveyor? No, all of the wheels are free spinning so all of the wheels double their speed, and the plane continues to move with respect to the ground. We know this to be true because the lentgh of the airplane and the distance between the wheels also remains constant, and if the plane remained stationary on the tread mill we would have a problem. We have a problem because the front and rear wheels are all free spinning, they do not provide the forward motion of the plane - they just spin. If your argument is correct that the plane remains staionary with respect to a point on solid ground while on the tread mill and if the front wheels are on the treadmill and the rear wheels are on solid ground, then the front wheels of the plane on the treadmill have a counter motion and so the result can't be greater than zero, so there is no forward motion; yet the rear wheels (on the solid ground) have no counter motion, so those wheels are moving forward...But how can the rear wheels be moving forward if the front wheels are not moving forward when the distance between the wheels remains constant? That is physically impossible!

In fact the front wheels of the plane do not remain stationary in relation to the ground, they move in relation to the ground. That is because the rear wheels are moving in relation to the ground (otherwise planes wouldn't be able to take off of a normal runway) and because the distance between the wheels remains constant the front wheels would have to move forward in relation to the ground on the conveyor to maintain the constant distance. Now (this is important) unlike a car, the front and rear wheels of a plane are all the same in that they are free spinning and do not create the forward motion of the plane, they mearly react to the forward motion of the plane. Since the free spinning front wheels move in relation to the ground on the conveyor this means the rear wheels (being constructed identically) will also move forward in relation to the ground on the conveyor. That means, unlike the wheel driven cars, the conveyor does not cancel out the forward motion of the plane! That means with all the wheels on the conveyor the plane will move forward in relation to the ground, even if the plane starts with all of the wheels on the conveyor.

Since the plane is moving in relation to the ground on the conveyor, then the plane has a groundspeed and a windspeed, and does in fact take off.

The only difference between the plane on a tread mill and the plane on solid ground is that the wheels on the treadmill plane are spinning at twice the rotational speed of the wheels on the solid ground plane.