Oh oh, the crowd is starting to get ornery now.

I'd better give the answer before things get worse.
The problem was a sticking solenoid. On the 4310 there is a solenoid/hydraulic valve that engages the PTO clutch. I ran two wires (one fused) from the battery to the solenoid and it wouldn't click. That was my clue. I kept trying power on, power off, over and over and it started clicking a little, and then more until after a few more times, it sounded healthy again. I hooked up the connector to it, and it worked perfectly. Talk about a lucky break.
One comment about the shop manual - In their diagnosis procedure they have you start at the power source and measure voltages at a whole bunch of different points on various relays and switches that you have to hunt for and identify. Eventually, you eliminate every possible problem until you end up at the solenoid. In my case, I would have been way better off to start with the solenoid. I guess it could have gone the other way, though.
Thanks, guys, for playing along with me on this. I hope it was at least a little bit interesting for you.
Tom