...Which means the wheel must turn faster. Which turns the prop faster, which turns the wheel faster. Why does that not work in still air?
Push the cart, and it sees a headwind, which you say is tailwind to the spinning prop, so off it goes?
As long as....
What wind pushing from the rear? There is no tailwind. That is why dwfttw is considered to be impossible. You have the cart generating it's own headwind and tailwind! Good grief.
It would not work in still air for the same reason a land yacht would not work.
It is not generating its own tailwind; the wind is the tailwind. It is generating a "sail" surface which is travelling below windspeed.
Imagine a more conventional land yacht, for a moment: No problem with something like that being pushed directly downwind, is there?
Now imagine it had a moveable mast which could slide to any position along the full length of the yacht. Silly, but it would still work just like a yacht.
Now imagine a mechanism to drive the mast along its slide, geared to the yacht's wheels.
You might want to imagine a clutch too, so that the mechanism is initially disengaged.
With the mast in its forward position, run the yacht DDW to its top speed.
Now engage the clutch.
The mast moves backwards.
As it moves, the windspeed from the sail's point of view has increased.
The yacht begins to speed up a little.
Unfortunately, it quickly runs out of travel and the yacht will resume its former top speed, now with the sail at the back.
(Take a breather there and please indicate if you have any objection in principle to the mechanics of the above)
Next, if you don't object, we can start on how a propeller blade presents the equivalent of a constantly-retreating face to the air behind it.