I'm not sure. I have to think about it some more.
The two winds pushing against each other doesn't sound right, somehow. It reminds me of the argument that rockets can't work in space because they have no air to push against.
My general way of thinking about the problem is something like this, although I haven't worked out the details: Suppose the cart is moving at the same speed as the air. Consider all the forces on the cart. The air exerts no force, directly (apart from the spinning propeller, which we'll get to). The ground exerts a backward force at the wheels. The ground also causes the wheels to turn, which spins the propeller. The spinning propeller exerts a backward force on the air, and thus a forward force on the cart. So, in order for the cart to speed up, the forward force from the spinning propeller needs to be greater than the backward force from the ground on the wheels. The magnitude of that backward force depends on how hard the wheels are to turn, which depends on how hard the propeller is to turn. If the propeller is relatively easy to turn, yet gives a relatively large forward thrust, the cart will speed up, and end up going faster than the wind.