The case with the wheels is a bit different from the prop case, but I think it contains the same difficulty in understanding. As an advantage the basics of mechanics are a bit simpler and less of an approximation. So it may be OK to look again at this simpler picture for the start.
To judge if the equations are correct or not the first thing is to make clear what is meant with P_out and P_net mean and the condition for the vehicle.
For the condition of the vehicle the situation to look at, the most useful case would be the vehicle standing still in the picture and with F_G = F_M to have a stationary case with no acceleration.
So what do P_out and P_net stand for ?
The equations are for the vehicle as the reference so vehicle is considered fixed not moving.
Pout is the output on the Motor wheel. Each of the two wheels is on a different independent medium but the wheels can not move relative to each other as they are connected by the body of the vehicle.
So taking case A.
If you want to take 10W from the front generator wheel then you will end up with a 5N force trying to push vehicle backwards right to left. But you can not have this 5N force unless you also have a 5N equal and opposite force and since this vehicle has just two points of contact the only place that equal and opposite force can be applied is back motor wheel against the Wind treadmill.
Pout will for ideal case contain Pin since all power generated to Generator wheel will be delivered with no losses to Motor wheel but to that power there is also the power that Wind treadmill will apply to motor wheel that will be the same force 5N that already exists at M wheel times the speed of the wind treadmill witch is wind speed minus vehicle speed.
Since Pin has a different direction than Pout the Pnet shows the net value.
If you will want to make calculation of speed maybe plot a speed over time graph then you can select either a speed step or a time step to do the calculation.
Say you select a time step of 1ms then you will calculate the kinetic energy of the vehicle after 1ms of 20W net power applied to vehicle and so vehicle kinetic energy will be whatever it was before + 20mWs (20mJ) and then from kinetic energy you can calculate the new speed of the vehicle at t+1ms and use those new values to calculate the new Pnet and then again add that to existing kinetic energy then again calculate the new speed and this way you can plot a speed vs time graph.
The main point of this 3 example's was to show that when you exceed wind speed the direction for wind treadmill has changed and thus instead of helping you to accelerate it will help you decelerate.
To get around this and ignore the energy storage Derek just decided to reverse the equation and use vehicle speed minus wind speed instead of the correct one.
You can not just change the equation when you exceed wind speed and same equation will need to be used for all 3 cases. If Derek was to test his equation for vehicle below wind speed case A he will notice that result will show vehicle deceleration and that will make no sense as that is not what happens in real world. Blackbird is many times pushed started but that is not needed it can start without being pushed.