I could think of nothing else as I rode home and I’ll concede the error in mty previous theories of wind gradient. (Dynamic soaring remains lots of fun..).

The Thin Air explanation seems to have it.
Doing away with calculus because I’ll confuse myself, a non-rigorous power explanation at least satisfies me.
Say the car is already moving at the windspeed V so there is zero relative windspeed. A force f at speed V extracted from the wheels as a propeller drive would tend to slow the vehicle and provide extracted power fV. In order to maintain the vehicle speed and ideally make it go faster, a larger force F must be applied by the prop. This power The prop applies it’s available power relative to the local airspeed (currently zero). So it’s easy for extracted power fV to apply a large push force F against the (already moving) air because v is small, and locally tending to~0.

So F>f and so overall the vehicle will accelerate beyond windspeed until they are equal.

QED? well it satisfies me anyhow.

ok, Here is the best explanation i have seen

Start with a vehicle where the spinning of wheels on the ground drive the propeller which moves the machine. Let’s assume, at first, that it is a perfectly frictionless machine, perfectly aerodynamic, with 100% efficiency in the propeller.

In dead air, if you pushed it forward, it would continue to move forward at exactly the speed you pushed it because the wheels drive the propeller which keeps it moving.

If there was a slight tailwind, then it would continue to accelerate. The wheels would always be pushing the propeller a tiny bit faster than what is needed for the speed it is currently going, because the air speed is slightly less than the ground speed. Suppose the tailwind is 1 kph. When the wheels are rolling at 2 kph, they are spinning the propeller fast enough to go 2 kph, but that is 2 kph relative to the air around it. Since the air is still moving at 1 kph, that means the propeller is trying to drive the vehicle at 3 kph relative to the ground. Once you reach that speed, the wheels are now pushing the propeller enough to go 3 kph relative to the wind, which means 4 kph relative to the ground. etc.

Now re-introduce reality. There are four energy drains to overcome: First is the internal friction of the vehicle, which can be made to be very small with good engineering. Second is the wind resistance, which is not significant until you have at least beaten the challenge of exceeding the wind speed, and can be reduced with good aerodynamics. Third is the internal efficiency of the transfer of energy from the wheels to the propeller (which their blog says was the biggest challenge). Finally (though they don’t mention it), there is the need to salvage all the energy from the ground speed. If the wheels start to slip, for instance, you don’t get all the energy pushed back into the propeller. They had to introduce friction here, but it is rolling friction, which is pretty tiny.