The schematic shows oil going over the high pressure relief flows from the high pressure side to the low pressure side. Since it is not dumping to tank, the relief is not the cause of your dropping charge pressure. The outside of your cooler looks sad, but that is not an indicator of what it looks inside. You are getting charge pressure so the circuit looks intact, but you don't know how much charge flow. I didn't read everything thoroughly but from what I did read, I didn't see anything about what the charge pump output should be. If the charge pump isn't putting out enough oil to make up for leakage in the high pressure loop under load, that would cause your pressure drop. Have you tried bypassing the cooler altogether to see if there is a restriction? I see the pictures of the piston slippers but couldn't identify if your pictures are of both the pump rotating group and motor rotating group. Are both shown in the picture?
All of the symptoms point through wear in one or more of the charge pump, hydro pump, or hydro motor. A piston pump like your hydro needs to be pressure fed or nasty things will happen like slippers pulling off the swash plate. If it has done much of this you should see things like erosion on the swash plate.
The feed valve/neutral valves on your system also show paths to the sump but I don't read much about them to know if the paths that bleed off leakage past these valves are large enough to caus loss of all your charge flow. It doesn't seem likely they are a problem though because your system works okay when it is cold.
Don't suppose there is someway you can beg, borrow or steal a flow meter to put in the charge circuit to see how much flow the pump is putting out?
