So i googled john deere 2130 pump and went to image search and found this. There's a lot i
cant tell about how it works from this, but i CAN tell that yes, the front pump is sucking the same fluid as the trans pump. It looks to me like the 'trans pump' is basically the 'charge pump' for the big pump up front. A low pressure pump sucking fluid through a strainer, pushing it through a filter, across a hydraulic block thing where you can access/test the various flows/pressures, on to the high pressure pump in front, which splits the flow: low pressure flow to pass through a cooling loop and serve as low pressure lube and charge for other things like maybe an engine governor and the brakes, shaft lube in trans etc. The other flow is the high pressure which tees off to the steering box, then goes to a pressure relief valve which tees off again to feed the 3pt and the external hydraulic ports. That's my theory from this picture, anyway.
So where are you finding the contaminated fluid? If you're finding it under the #15 cap in the picture, it had to go through 2 filters to get there. The inlet strainer for the trans pump, and the 'main' filter after that. The inlet strainer may not be easy to access but if it has literally fallen apart it could be that the trans pumps is eating chunks of it and spitting them out (and throwing off its own debris in the process). If the inlet strainer is functioning, then in theory the 2nd, tighter filter #12 should catch most debris that the inlet filter didn't, but it does look like it has a bypass so if it's clogged stuff could be going right past that filter through the bypass.
So it kind of depends on the size of particles you're looking at. Bigger heavier stuff shouldn't make it past the first strainer, but the strainer could be failed. If the strainer is good, the trans pump could be grinding up and making debris, but that should be caught by the 2nd filter. If the 2nd filter is clogged and bypassing, or just falling apart, debris from the trans pump, or the filter itself, could be going through the front pump and then ending up everywhere in the system.
Or if its small enough stuff that those filters wouldn't catch it in the first place, then your dealer is right and it could literally be coming from anywhere and you'd have to do a huge teardown just to find out the root cause. So i guess the critical question is:
Can these particles fit through either or both of the filters in the system? Depending on the answer to that, it either narrows it down to the relatively short path leading to the front pump inlet, OR... it makes it so wide open it could be basically anything in the whole **** tractor.