I initially thought that the direction of the propeller was set so that a tailwind would make it rotate in the direction that makes the wheels go forward. Only after the treadmill explanation I realised that it's the other way around.
The direction of the rotation of the propeller in response to tailwind depends on whether the device is on the ground or whether you hold it in your hand. In the former case, it rotates to make the wheels go forward, in the latter case, it rotates to make the wheels go backward.
So the initial acceleration is not caused by the rotation of the propeller, but by the drag on the entire cart.
That drag helps, but it's actually the air pushing on the propeller that accelerates the vehicle.
As the cart speeds up the propeller provides less and less drag to the wheels and at some point starts providing net thrust.
The propeller provides net forward thrust to the vehicle at all times.