J13, I think we are almostin agreement, but not that the actual device will work.
I would like to stick to the demonstrations for the moment if I may, because here lies a source of confusion.
Spork, try this (thought) experiment. Place the cart so that is runs in the same direction as the belt. Well, it travels at the speed of the belt or perhaps a little backwards. Now hold it with your finger, and allow the propeller to turn, and then release it. It will now move faster than the belt. This has nothing to do with the propeller, because in this direction, the same effect can be achieved with a mass as with the propeller. If you impede the cart briefly, so that it has less opportunity to pick up energy, it will come to a stop before the end of the belt, when the mass (or propeller) stops. So would it the other way, given a chance.
In the other direction, it travels at (almost) the speed of the belt because the propeller itself creates drag. Just like you need a finger to spin the propeller in the first place. Drag 'sticks' the cart to the belt. The propeller doesn't drive the cart, it couples it to the air, so that the belt can drive it. If a momentum-less propeller were possible, the cart would drift backwards, because 100% transfer is impossible, but the initially stored momentum makes up that loss and some, propelling it forward.
Only the stored momentum does work in both cases. Even the sliding will be enough for just a little bit more energy be gained to keep it running. Sorry, but it's an illusion.
In your demonstrations, you are seeing the movie, but not the beginning and the end. Momentum is always conserved. The system must includes all sources of energy, including your input, and run continuously. Many over-energy devices (I know this is not your claim) appear to work when you see only some of the inputs, or for a short time.
The real cart is also a matter of momentum but for another reason. I could try to explain...
My_wan,
It's not skepticism, but physics. This problem is merely the result of intuitive thinking. No shame in that, it led Da Vinci to make many errors.