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: 
The comparison with a car's gearbox is unfortunate, and leads to misunderstandings...

At the small airspeeds typical of takeoff, the inflow at the rotor plane is small too, and the prop needs a small amount of pitch in order to produce a large amount of thrust at maximum engine revs, the blades working at the angle of attack for best L/D.

But, on cruising flight, there is an important inflow at the plane of the prop disk, that inflow reduces the angle of attack, and –under those conditions– the propeller can deliver enough thrust at maximum engine revs (i.e., with the blades working at the angle of attack for best L/D) only if the pitch is substantially higher than at takeoff...

That's the rationale of the adjustable prop...
### 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.