italia:
Regarding your parcel of air explanation: as discussed, yes, it requires power to move air. That power comes from the engine. That power is called BHP or SHP. It is not called THP.
The power that comes from the engine goes through the prop. Rudderrudderrat is correct about propeller efficiency not being zero. If it was zero there would be no thrust.
You can't work out the prop efficiency in the static thrust scenario using the formula you keep giving for propulsive efficiency. I have found an essay by a prop designer that explains why:
Theoretical Prop Efficiency
"This equation [propulsive efficiency] is very useful for many cases, but you should see a problem in that as your velocity goes to zero, no matter how much thrust you're producing, your efficiency goes to zero. So how do you know how good your prop is doing at low speeds or statically? Well, there's another term that can be used, Figure of Merit, which is Induced Power divided by Power Available, or how much of your power is going into accelerating the air..."
Induced power is the same as thrust horsepower.