I think a possible source of confusion is talking about "wind" instead of talking about "speed of the air". Saying "there isn‘t any wind relative to the vehicle" is equivalent to saying "the air is not moving with respect to the vehicle". This does not mean that the air no longer has any effect, or cannot be considered as a factor in the equation: the vehicle is still in contact with the air, so it can push against it.
The movement of the propeller is indeed created by by the motion of the rolling treadmill surface relative to the vehicle, but this motion only continues because the propeller is thrusting against the air. Imagine what would happen if the propeller was not connected to the wheels: in this case, when the vehicle was lowered on the treadmill, the wheels would start spinning, but the prop wouldn't turn. The loss of energy due to friction would soon cause the wheels to spin slower and the vehicle would run off the left end of the treadmill (viewed as in Spork's YouTube videos). In fact, when the propeller is connected to the movement of the wheels, it thrusts against the air causing the vehicle to run off the other end of the treadmill.
The essential thing is that it's not "the wind" that is the motive force, it's the difference in speed between the air and the ground. If there's no difference, the vehicle won't run. If you go out on a day with no wind at all and put the vehicle on the ground, it won't do anything. If you now give it a push, there will indeed be a "wind" relative to the vehicle, but it won't help, since there is still no difference in speed between the air and the ground. In this case, the vehicle will run a certain distance but will be quickly slowed down by forces of drag and friction.
If there is a steady wind, the vehicle will accelerate up to the speed where the forces are in equilibrium, then just keep running. And it's important to point out that the point where the forces are in equilibrium is not necessarily the point where the vehicle is stationary with respect to the air. When it is stationary with respect to the air, its wheels are being turned by the ground, the propeller is being turned by the wheels and therefore thrusting against the air.
If the vehicle is going at exactly the speed of the wind, then all the extra thrust of the propeller is doing is compensating for any speed losses caused by drag or friction. If the vehicle is going at 10mph in a 7mph wind, then we might consider that the thrust of the propeller is accounting for the extra 3mph. It's a simplification, but it may be a step towards understanding what's going on.