Okay. With a herculean effort, and a few new tools, I was able to lift the spindle assembly off the top, remove the split taper hub, and pulley. I kept disassembling and found I was able to remove the lock nut on the Z-Axis ball-screw with hardly any effort. I did it with my bear hands without even holding the ball-screw in place. In other words, the lock nut was super loose. Upon further inspection and a little googling, it became clear that the lock washer was supposed to have (at least) one of its tabs folded up into the lock nut to keep it from coming loose. This either wasn't done, or the tab that was bent up had since broken off. I think that explains the backlash I was seeing. I was happy to see that the bearings appear to be in good shape.
So the repair seems pretty straightforward at this point: tighten the lock nut back down, fold up one of the remaining lock washer tabs and put the whole thing back together.
My question now (mikecolee, maybe you'd be so kind as to see if the manual is any help on this) is how tight do I need to make it? Is there a torque rating? Or just hand tight? I'd like to avoid guessing, putting it all back together, and finding I guessed wrong. Even if I guess, I'm sure the machine will be tons better than it has been. I'm very excited to have it usable again!
Thanks again for all your help!