In my mind, there are a few possible causes of no flow to the loader / 3pt, either intermittant or otherwise.
A blockage / or restriction in the pressure side of the system that prevents the fluid from getting there. I think we can rule this out, because it would load down the pump and engine, and should be noticeable.
A failing relief valve that is sticking open and allowing the fluid to flow unrestricted back to the resevoir, thereby providing fluid but no pressure in the circuit, and no ability to do work. Should hear it, but you never know.
A pump problem where it is failing to put out pressure due to an internal failure / blown O-ring / leakage due to wear.
A problem in the suction side of the circuit that prevents the pump getting fluid to pump. Could be not enough fluid, a blocked or collapsed suction line or screen, or air leakage.
You've already eliminated a few of the above. The engine is not loading down, therefore there is no restriction in the system. You checked the screen and changed the fluids, so I rule out not enough fluid / wrong fluid.
I'd start with the relief valve as it is an easy check. If it is sticking open, then you will have all the fluid going directly back to the sump. Not sure why 20 minutes would affect that though - it would be more likely affected by hitting relief pressure using the loader.
Second, I'd be looking at installing a pressure gauge in the line upstream of the relief valve to try to see if there is any pressure / flow at all when the problem starts. No pressure at all - no flow - some pressure you have flow but it's going back to the sump.
Third I'd go back to the suction side to see if there is a reason that the pump can't prime. If you have something floating around in the system that can block an inlet, it would manifest itself this way. Works fine until it gets sucked over the inlet port, then no flow until the pump stops trying to suck and it can drop back off. An air leak would more likely be an immediate problem.
Finally I'd look at tearing down the pump or replacing it. However, if you have access to a spare one you could borrow, I'd swap them a lot sooner to rule that out.
good luck.