Does a constant power prop exist? What are the tradeoffs compared to a constant speed prop?
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.