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.
Thats why I said
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. 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).
With the throttle open you will not get shock cooling from the extra airspeed
As for the pitch stability, I have not noticed this but I usually trim for the descent anyway so this may mask the effect.
£2-3 per hour is a good estimate if you don't have anything go wrong and fly a lot of hours. It can be more than this is you fly lower hours as the 3 yr overhaul tends to be a fixed cost and therefore the hourly rate depends on the hrs flown. If you need a new prop then a 3 bladed for a Piper Arrow is about £8000. We needed a new one because the hub was said to have corrosion, although we couldn't see it. I don't think fixed pitch props generally get hub corrosion.
You are correct that the benefits generally outweigh the costs but I was just pointing out that there is a small negative to offset the advantages.