Tell me: From a frame of reference travelling with the air (or standing still with it), what is the kinetic energy of the cart in both cases?
In the real world 1/2mv2, where V is relative to the ground, in both cases, because the treadmill is also in the real world. It is going as fast
as it appears to be.
And from a frame of reference fixed to the road or treadmill surface?
The wind is indeed necessary, but only insofar that it creates a difference in speed between the surface in contact with the wheels and the air in contact with the propeller.
Only the ground is relevant. It is common to the roadside and treadmill side observer. You could do it from the belt, but that will not mean any effective change in any parameter. If you find a real moving road then perhaps it may be useful to know from that vantage point, but why consider it?
Velocity in that last case cannot be used to calculate power for example.
It means nothing without knowing the force as well. Low friction in this case, means low force, and it is clearly low
There is a word game here. Equivalency means for example, that if you view on object from another, they remain in the same relative locations. Also equivalency means, that unless velocities are very high, all objects are unchanged relative to each other by their velocities. There are no Einstein effects. Two parked cars may have a lot of KE from traveling around the sun, but they, like all objects on this planet have it too, so the relaative difference is zero. That is our reference frame. There is nothing to be gained from making the observation that one can be viewed equally from the other. So what?
This has nothing to do with the problems of the treadmill. It is a distraction.
Imagine a cart in a 20kph wind but on a belt. When the beltspeed is zero, that is like a fixed road. The only way that the cart can be at 0V relative to a ground observer, is if the belt and wind are equal and opposite. That is, the belt is going back as in the treadmill, at 10kph. Two opposing velocities mean that the cart is stationary.
This is also true for the observer in the car. In the real world, that is the only case, save for no motion at all, where these two views are the same
How is it that the treadmill can model a situation, that of being at windspeed, but stationary to a ground observer and the driver?
There is no relative velocity between you and the driver. When that happens it is not possible to be moving. The model is wrong
Only the boundary layer is modeled, and this is insufficient to couple the cart to the belt in a useful way. in fact, it tends to lift the wheels away, which is why the cart appears to float. What should happen, is the cart should go with the belt, dragged back by the missing wind. The cart would need to drive against that to achieve windspeed.
The entire model is wrong, because there is no real world equivalent that it could model.
You are seeing an artifact that results in the balance I have mentioned, that lends the illusion that the cart is at windspeed. Without that, you would not give it a second thought.