Question: From what I understand, the purpose of a variable-pitch propeller (be it manually adjustable, or constant speed) is that it allows the engine to always operate at its most efficient speed, much like the transmission of a car. Is that correct? If so, if an airplane were to accomplish that another way (say, a hybrid gas-electric drive, or an actual transmission with shiftable gears\*), would there be any other advantage to having an adjustable propeller?

\*No, I don't know **why** you would want to mount a car transmission on an airplane. It's a hypothetical question. But now I have this picture in my head of a twin-engine plane with two gear shifts in the cockpit and two clutch pedals next to the rudder pedals.

Answer: 
It's not just the operating RPM that is pertinent. It's also the pitch of the blades.

If your propeller blades' pitch is optimized for takeoff/climb performance, it's going to be aerodynamically inefficient at high true airspeeds, like taking little tiny footsteps on a treadmill that's going really fast. (It works, but it wastes energy.)

The opposite is true as well. If you have a "cruise" prop on your airplane, it's going to perform poorly on takeoff when the airspeed is slow, like taking large strides on a treadmill that's just barely moving.

If you want to optimize your body mechanics, you change your gait based on the speed you need to go, so you're not wasting energy. If you want optimize a propeller, you change it's pitch based on the speed of the air that's entering it, so you're not wasting energy.
### Reverse Pitch - Reverse Thrust

If the pitch angle is negative (reverse), then some meaningful amount of thrust is produced in the opposite direction than normal. This can be useful to improve stopping distance when landing, and even taxi in reverse.
[https://www.skybrary.aero/index.php/Reverse\_Pitch#:~:text=When%20installed%2C%20reverse%20pitch%20is,of%20motion%20of%20the%20aircraft](https://www.skybrary.aero/index.php/Reverse_Pitch#:%7E:text=When%20installed%2C%20reverse%20pitch%20is,of%20motion%20of%20the%20aircraft).

### Zero Pitch - Idling

At somewhere around a pitch angle of zero, the net thrust becomes zero (forward or reverse). At this point the aerodynamic load on the prop is minimized. This removes unwanted thrust when we want to stand still on the runway, as well as minimizes fuel consumption during idle.