That's why propellers have angled surfaces.
To get from the ice boats to the downwind cart without woo, just follow these simple steps.
Step 1 is to acknowledge that ice boats can not only sail at speeds faster than the wind speed, they can sail steady-state (without exploiting gusts, momentum from turns, etc) such that the downwind component of their velocity is faster than the wind. This can be verified from ice-sailing sources.
Once that's understood, all that's needed is to transform an ice boat into the downwind "cart" in a series of steps, each of which preserves the essential mechanics of the scenario.
So, step 1 is an iceboat sailing at a downwind angle in a steady wind on an infinite ice surface. The directly downwind component of its velocity is faster than the wind; it's speed of course is faster still.
In step 2, wrap the ice surface into a tube (larger in radius than the height of the iceboat's mast), with the iceboat inside. The wind flows continuously down the tube; the boat on its slanted downwind course now sails a helix in the tube. (Ignore gravity; if you must, assume zero gravity and that the centrifugal force of the spiral course is sufficient to keep the boat's blades pressed against the ice enough for them to work as normal.)
In step 3, add a second ice boat. It is identical to the first, and it sailing the same way, but it is positioned 180 degrees around the tube from the first one. If either ice sailor looks straight "up" he's looking straight down on the other boat.
In step 4, lengthen the two boats' masts so that they connect in the center, rigidly attaching the two boats together.
In step 5, add a threaded rod running down the center axis of the ice tube. This rod passes through a hole in the two masts where they join in the center. The threads have the same pitch (number of turns per meter) as the helical path the ice boats are sailing along.
In step 6, add screw threads to the hole in the masts, that match the threaded rod. Since the hole threads and the rod threads have the same pitch as the path the the boats were traveling anyhow, this will not (aside from adding a bit of friction which we can ignore for the moment, for reasons explained in the next step) alter the courses or impede the movement of the boats.
In step 7, we remove the runners from the ice boats. They no longer contact the ice. However, the screw threads serve the same function that the runners used to, of keeping the boats' course on the most efficient angle. There is no longer any runner friction, but the friction of the central bearing replaces it. Since in principle either can be minimized, call it even.
In step 8, we remove the ice tube, since it no longer plays any part (except maybe to contain the wind in a zero gravity environment, so if it helps, assume the tube is replaced with a much larger tube, still containing the same wind velocity along it.)
In step 9, we introduce a long strip of flat pavement parallel to the tube. The pavement is far enough from the threaded rod that the two boats stay about half a meter away from it at their closest approach. At the same time, we also re-introduce a gravitational field that attracts the boats toward that pavement. Since the boats are still held "up" by the threaded rod, and they counter-balance one another, this does not affect their motion.
In step 10 we add a three-wheeled cart that is in contact with the pavement. We also add some struts that extend from the cart to a bearing around the central threaded rod. The bearing where the threaded rod goes through the boats' masts pushes against the cart's bearing, pushing the cart along at the same downwind speed as the boats.
In step 11, we remove the hulls of the boats, since the pilots can now ride in the cart instead.
In step 12, we remove the threaded rod. Instead, we add linkages and gears from the main axle of the cart to the bearing around which the boats' masts rotate, with the gearing arranged so that the angle that the boats' mast rotates for a given rotation angle of the cart's wheels is the same as it was before.
And we now have the downwind cart. Note that from the start to the finish, the fundamental device that makes it work remains the same: a surface moving laterally at an angle to the wind, whose downwind movement is constrained to maintain a certain ratio to its lateral movement. At the start the surface is an iceboat sail and at the end it's a propeller blade. At the start the mechanism that enforces the ratio of lateral to downwind movement is the iceboat's blades and at the end it's the gearing between the wheels and the propeller. But the basic dynamics stay the same.
If you still have difficult accepting this, please point out in which step I introduced the woo.
Respectfully,
Myriad
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He's not still trying to claim that balloons can't travel at wind speed because friction through the air slows them down, is he? 
