Personally I would start back at square one trying to troubleshoot the problem. I've owned two JD 750's and the Yanmar engines on those were pretty similar to the one on your 755.
My first step would be to PULL the battery out, take it somewhere and get it tested. I would probably give it a good charge regardless whether you think it needs it or not. A fully charged battery should read about 13.8VDC with no load. So, when you put it back in and IF it does the same thing, then you have eliminated the battery as the cause of the problem.
Next I would double check all the major connection points on your heavy cables, both positive and ground wires, plus all the lighter gauge wiring to/from the solenoid and the ignition switch. Thoroughly clean as many connections as you can see while you are looking them over. If you see any in-line connectors, pop them loose and have a good look at the blades/pins. Sometimes they can get pretty corroded if a contaminate gets inside it. If the engine still won't crank, then you have pretty much eliminated the wiring and connections (with one exception I will get into later on).
The fact that you can hear the solenoid clicking tells me that most likely all of the safety interlocks are working correctly. If you heard no clicking when you turn the key, then I would say you would need to investigate all the safety switches/interlocks at this point as the potential cause, but that doesn't seem to be necessary in this case.
Since the solenoid will click when you turn the ignition key on to crank it, I think you can eliminate the ignition keyswitch. IF you heard nothing click when you turn it, then at this point you should measure the contacts on the switch with an ohmeter to see if they are closing. Since the contacts seem to be closing in your ignition switch, I think you can eliminate that as the cause of the problem.
That's getting down to just a few things left.

Before you do anything else, try this: Turn on the ignition key and HIT the STARTER MOTOR solidly with a heavy rubber mallet with the keyswitch on, or maybe use a small hammer. Just don’t hit it hard enough to break the cast housing. If the starter motor magically spins over, then you have the same problem I had on my 750.
You're correct that the solenoid is integral with the starter motor. I'm going by memory here cause I had to pull the starter off of my 2nd 750 and tear into it. It turned out to be ok actually, the starting problem I had with that tractor was an altogether different issue that I will go into in some detail below. If you must pull the starter off, then do it. It's not that hard, if I can do it I'm sure you can!!
The contacts are inside it on the end of the starter opposite of the reduction gear. When the coil is energized, the plunger slams the contacts together completing the heavy current path to the starter motor. Check these contacts, clean if needed. IF that’s working ok, and the contacts are in good shape, then its looking like the problem could be the starter motor itself. You can benchtest the starter with a spare battery, or a battery charger, just to see how well it runs by making the starter motor spin over while sitting on the bench. It's DC so it can spin either way, depending on how you connect it to the power supply. But, to be safe, you should connect it + to + and - to -. Also, watch for the motor to jump when you put 13 volts on it, it has a LOT of torque and it can jump off the bench if not clamped down good.
Anyway, to get to my last point the guy at the starter shop had seen my problem before. The solenoid on mine would click, but not turn the starter over (mine was doing the same thing yours seems to be doing). I told him I could bump the starter with a hammer (with the ignition keyswitch in the crank position) and it would spin over. He fixed me up an extra relay, told me how to wire it to the solenoid, and said it would fix it. Well, it worked like a charm and it cranked fine everytime after that. I’ve kept in touch with the current owner of that 750 and he says it still cranks with no problems whatsoever.
This problem was fairly common and is the result of some design flaws that are also seen in other tractors. It's a combination of wiring that was undersized, and poor connections over a period of time. To be more precise, some of the wiring on a good many JD tractors should have been made a bit larger. The undersized wire will increase the resistance of the wire and at the same time increase the voltage that is dropped across the length of the wire. Also, over the years the wiring connections can work loose, or just build up some slight resistance from light corrosion. On my tractor, these voltage drops (both in the wiring and in the connections) combined enough to decrease the voltage that was getting to the solenoid and starter. This would not let the solenoid slam hard enough, nor would the starter motor have sufficient current to spin the diesel motor over. Remember, this wiring resistance and any resistance in connections will drop the voltage slightly and at the same time will limit current flow to the components it's going to! When you’re cranking a cold diesel motor in the winter, you need EVERY amp you can get to the starter.
Hopefully by now you will have found out what's wrong with it! Good luck and let us know.