I have not had a boat (or even operated one) with stabilizers. But I have worked on a fair number of similar hydraulic systems on vehicles, tractors, and construction equipment over the years - diagnosing problems, rebuilding pumps, hoses, racks, etc.
1) I interpreted - and maybe I was wrong - Martin's comments to imply the pulley shaft was overloaded in this application, and the noise might not be the belt itself, but accumulated wear on the shaft or bearings in the pump. So once it's worn enough to whine, a new belt isn't going to change things, and only rebuilding the pump and re-engineering the pulley so it didn't wear as quickly again. Or maybe I misunderstood his point.
2) You can isolate the noise using an improvised stethoscope (or a real one). A large screwdriver is adequate, or any solid rod like that. Put one end on various parts of the system (pump, at the stabilizer, etc.), the other on/next to your ear against bone. You will hear the noise transmitted through the system, but when you are at or close to the problem it will be very pronounced and obvious. This can neck down your problem considerably.
3) Again, I know nothing about the Naiad system in detail. I tried to find a diagram of the hydraulic circuit from a google search but oddly, no success. Maybe something is unique to this system, but it sounds really unlikely or at minimum weird to me there would be a blockage develop in the low-pressure part of the system between the reservoir and the pump. That is, overall, the very least stressed part of any hydraulic system.
4) Using some physical reasoning and your comment that the whine takes a few minutes to develop, it leads me to put wear and not blockage/constriction as a higher probability in a differential diagnosis. I say this because the pressure in the system is consistent after the first few revolutions of the pump onward. A cavitation/starvation issue should be apparent right away. What does change over a few minutes throughout the system is temperature. This has two main effects a) greatly thinning the fluid, and b) increasing tolerances in components from thermal expansion. Either of these, and especially the combination, can create a noise as metal now contacts metal unlike when cold. Of the P,V, and T in a fluid or gas system, it seem to me that since P and V (assuming the system is full and not leaking or you would have said so) have no effect on the noise, and T appears, to, I think about what that implies.
As I said, I've worked a bit on similar systems and PTOs, but all are unique and I don't know this system.