John:
It sounds like your tractor uses the transmission case for a hydraulic reservoir (the pump may be in the transmission case or mounted on the engine with lines to the transmission case).
In either event, the spring loaded valve you mention must be a one-circuit selector valve that directs the pump outflow to one of three places. When it is in neutral, flow is directed to the three point hitch. When depressed, flow is directed to the FEL. When raised, flow is directed to the rear outlet.
What doesn't make sense is why you have to hold the selector valve in the depressed or raised position. Perhaps the valve has a detent that normally holds against the spring when you move it all the way to the limit. You may not have been pushing hard enough. Or perhaps the balls in the detents are worn or otherwise not functioning.
In any event, the previous owner must have used the Tee fitting on the transmission top as the return line for the rear remote connection. What doesn't make sense about that is why he didn't run a separate line from the tee to a companion quick disconnect so that both lines to the rear remote could quick disconnect. Perhaps it was just a quick make-do job (I have frequently been guilty of that myself).
In any event I think you are correct that the return line from the backhoe connects to the tee. If I were going to remove the backhoe from time to time, I would put a quick disconnect in that line as well.