I'm late and I like your explanation. Your Idea of freely spinning wheels is true, but remeber, there are power to the treadmill. So it's like an opposite upside down situation.

So it's like how your car is being dyno or when you do the OBDI Emission check. The Plane will be the rollers of the machine (don't know what's it called) and the Treadmill is the car itself.

However, your "real life" experience has nothing to do with the plane on the treadmill since

A. The paper doesn't work the same way as the tread mill. Treadmill puts out torque and power.
B. Plane requires horizontal air movement to lift it up.

Since:

1. PLane has +A velocity on +X Direction
2. Treadmill has -A Velocity on -X direction.

According to Physic,

If there's equal and opposite force that applied at the same time, there won't be any changes in the X direction movement.

Therefore, there won't be any -A^n velocity of the air movement.

The plane will not take off.