Does a constant power prop exist? What are the tradeoffs compared to a constant speed prop? Variable-pitch props allow pilots to adjust the angle of attack of the propeller to change the load on the engine.
The most common implementation of this is a constant speed prop where the prop governor adjusts pitch to maintain the chosen RPM.
From a pilot's perspective, we only really care about two things:

Power output
Fuel economy

Using a constant speed prop, you can't directly set the output power - you set the throttle and prop speed and observe the power output.
Using a constant speed prop, you also can't directly set the fuel flow rate - you set the throttle and prop speed and observe the fuel flow (or look up the POH).
Using a constant power control law instead the pilot could set the desired power and have the prop pitch adjusted automatically to maintain that (or if impossible, fail in a similar way to a constant speed prop going off the governor).
It is also possible to pick the most fuel efficient RPM for a given engine load which means that the two inputs (throttle + prop speed) get reduced to one: desired power.
The benefits I see are:

Reduced workload
Increased efficiency

The drawbacks I see are:

Inability to operate the engine at anything other than best efficiency for a given power - but I don't actually know what use case that would be useful in.
System complexity
Certification cost

Has a constant power control system ever been used in aviation? Are there other drawbacks that I'm missing?
There are some aircraft that indeed have just power lever and set the RPM on schedule, e.g. the Pilatus PC-7, PC-9, PC-21 line of trainers. Since they are primarily designed for training future jet pilots, and jet aircraft only have power levers, it makes sense to skip the complexity of separate propeller and condition controls.
However normally there are some cases where you do want to set other than most efficient RPM for current power and true airspeed:

The most common is descent and approach. These are flown with high RPM because

the engine is already spinning fast, so it does not have to spool up if you need to add power to go around and
with retarded power it produces more drag, allowing to expedite the descent when needed (the Pilatus PC-7/9/21 have a ventral speed brake; most turboprops don't).


Failure modes, though for most a feather switch would probably be sufficient.