Well, I started digging, even though I know better. I drained the trans before taking everything apart. Fluid looked OK, but smelled pretty funky. Then- I took the rear plate off. There was about 2" of semi-solid gunk in the bottom of it, about the consistency of grease. As I started taking it apart, I realized that it would all have to come apart to flush all of the crud out of there. I got the differential out, and the bearings are in horrible shape, deeply pitted. I assume moisture sat in there for some of the last 50+ years. Luckily, it's a standard bearing; $30 a pair on 'bay. The gears look fine; no evidence of rust or pitting, and no wear to speak of. I need to get the whole thing into a cleaning tank now and flush it out. I may have to replace the other bearings in the transmission, too, but we'll see. The outer axle bearings are in excellent shape; just need new seals.
I can see that this toy is going to take some time to do 'right'. It probably would have worked just fine the way it was, but pitted bearings rarely get better on their own.