Generally the first thing to make hydraulics stutter is when the pump sucks air because the fluid is running low.
The guy testing the pump would have noticed that, i hope.. Plus on larger machines we're talking many gallons of fluid that had to go somewhere, and it's usually not a mystery when gallons of fluids come out!
I could see some very tight/binding pivot joints cause the stuttering, but there's almost always popping, and squealing noises that sound like 'metal torture' when you're forcing such things to move.
I can't visualize exactly what might be happening because i don't know what hydraulic coupler guts look like, but i have a feeling the hydraulic coupler thing is either correct or directly related. It sounds like you have some kind of orifice created by a moving part in a hydraulic supply line that is shared by all the functions, which is moving/closing in response to flow, and when flow stops it opens up and operation resumes. If it does that over and over very fast it would feel like vibration. Hydraulic disconnects have moving parts that close the line and stop flow. If the couplers are fully engaged to each other those parts SHOULD be held open and have no response to the flow rate. But, i guess depending on the design of the coupler it might be possible for them to be locked together mechanically and not leaking fluid, but for the parts which are supposed to be held open to.. not be held open?
I have seen something similar with tire valves aka schrader valves.. The female side of the coupling is supposed to have a post/stud that pushes open the valve on the male side. If that post is bent aside or falls out for some reason, the valve will not 'flow' just because you've coupled to it.. but if you put enough air pressure to it it will push open the valve with air pressure and flow anyway. The valve is just held closed by a spring, so once the pressure pushing on the front of it is stronger than the spring, it opens. If pressure crosses above and below that 'threshold' repeatedly in a short time, the valve would 'chatter'. Dynamic forces from opening and closing control valves, or running the boom/bucket into dirt would cause those pressure variations just like water hammer from closing valves in water pipes can cause them to vibrate.
IF you had something like that going on with a hydraulic coupler, on an open center hydraulic system there would normally be almost no pressure until you operated a valve. But if this was happening your pump would always be making the pressure needed to 'crack that valve' in the coupler and this might create noise at the coupler and heat in the fluid. I would sure just try inspecting any couplers to the backhoe before digging in deeper.. Noise, heat, leaking, and if nothing else just reseat them and see if it goes away!
