Actually, if you're doing the analysis in the "ground's frame", it is force X distance over the ground.
What he is doing is mixing up energy required to do the work with energy balance. These are two separate issues.
In the first case I have 920 J/s KE loss of the air, plus 483 J/s of effective power spinning the perfect prop = 156 N X 9 m/s
I had lots of trouble understanding this, but now I've come up with an analogy that helps me. Can you tell me if it's truly analogous?
Consider a gas-electric hybrid car with regenerative braking. Imagine that, in addition to the gasoline engine, it has both a motor and a separate generator. Pressing the accelerator connects the front wheels to the motor, which discharges the battery as it drives these two wheels. Pressing the brake pedal connects the rear wheels to the generator, which charges the battery as it is driven by these two wheels.
Suppose the car is moving along at constant speed, and the gasoline engine is providing the power to overcome friction, with both the motor and generator inactive.
Now, the driver presses hard on both the accelerator and the brake pedal at the same time, engaging both motor and generator. Except for relatively small losses in the motor and generator, which are made up for by a correspondingly small input of energy from the engine, nothing happens. The car continues at constant speed.
If we focus on the front wheels, it looks like lots of work is being done on the car, because the car is moving forward and the front wheels are pushing the car forward with great force. Where does all this energy come from? The engine is only providing a small amount.
The answer is that it comes from the battery, which is getting it from the generator, the work to drive which is the very work whose energy source we're looking for. The energy doesn't need to come from anywhere else, like the gasoline engine.
Getting back now to the wind-powered cart, the moving air corresponds to the car's gasoline engine, which can provide some extra energy. The propeller, which drives the cart forward, corresponds to the car's front wheels and motor, which drive the car forward. The wheels of the cart, which are driven by the cart's motion, correspond to the car's rear wheels and generator, which are driven by the car's motion. The cart's drive chain, which transfers power from the wheels to the propeller, corresponds to the car's battery, which transfers power from the generator to the motor.
In conclusion, part of the power needed to do the (156 N)(9 m/s) of work per second which is being done on the cart can indeed come from the propeller, just as part of the power needed to run the car's motor can come from the generator. It doesn't all have to come from the wind. So, it is in fact correct to add the energy loss of the wind to the power provided by the propeller, to get the total work done on the cart by the 156 N of force on the propeller.