Oggers, some of your posts appear to indicate that you think that if thrust is not zero then propeller efficiency cannot be zero. If you do believe this, then you are wrong (once again).
If you do a search for the definition of propeller efficiency (I've just done one) you will find that the most commonly used definition is as follows:
Propeller Efficiency = (Thrust x Aircraft Velocity) / Propeller power input
If you look at that equation you will see that there are two possible conditions that will give zero propeller efficiency.
These two conditions are, when thrust = zero, and when aircraft velocity = zero.
If either one of these conditions exist then the propeller efficiency will be zero.
In your hovering aeroplane picture the aircraft velocity is zero, so the propeller efficiency is zero.
A number of your arguments appear to be based on the belief that the aircraft velocity is the same as the propwash velocity. It is not.