One other big advantage is in the descent. With a constant speed prop you just point the nose down and leave everything set. The engine cannot overspeed as the prop governor prevents it by coarsening the pitch.
What seems to have been forgotten is that a CS prop gives you superior pitch stability.
With a fixed pitch prop, if you point the nose down, the plane speeds up, the airflow through the prop goes up, the engine revs up like crazy (going from say 2300 to 2500 RPM in a few seconds) and (given that prop thrust is proportional to the cube of the RPM) this greatly increases the thrust, which causes the plane to speed up, which causes the wing lift to go up, which causes the plane to porpoise upwards. So holding altitude, or holding the glideslope on an ILS, can be fun.
With a VP prop, pitch changes done with the yoke don't have this effect of exxagerating the phugoid oscillation, resulting in much better pitch stability.
Because you don't throttle back you can usefully convert all your height to speed and there is no shock cooling (as the throttle is still open).
Not entirely correct; if you point the nose down, the airflow through the engine goes up, and the engine gets cooled more. Especially if done together with reducing engine power, this can excessively cool the engine. In fact probably the biggest engine management issues are in planes with the bigger engines (like an IO540 and upwards) which have CS props almost universally.
There is very little extra maintenance. The direct operating cost goes up by £2-3 per hour, which is about 1/20 of the fuel cost alone.