I have run into similar customer complaints in the past.
HST's don't generally fail in the manner you are describing. They generally fail in that the engine revs, the hst doesn't go anywhere or is very, very weak to move. And if that's the case usually the HST and hydraulic filters will be full of material that came out of the hst. So with that, if you suspect, take the filter off, cut it open and inspect it.
In the past on the GL 30 series I have messed with that had these complaints, it was almost always either an engine issue or a fuel issue. One time I can remember a hydraulic issue pulling power out of the engine which made one think that the transmission may have been an issue. Repair of the hydraulic system solved the customer's complaint.
Ideally you would diag with a dyno. If it makes rated horsepower on the dyno, the engine is not suspect. But that doesn't necessarily test the brakes, transmission etc. Yes a dragging brake can cause exactly what you describe.
Does it make any difference if it's in low medium or high range? You "ARE" using low or medium when climbing hills or pulling any ground engaging implements, right?
Do a compression test and compare it to the WSM's spec.
If compression is excellent, look into your fuel delivery system, this includes the nozzle assemblies as well as the pump. Before you pull the pump off/out have a look at the 2 governor springs. I've seen them broken in the past.