I've had the fan apart twice since my OP.
The oil reservoir was dry. I filled it and that helped a little. The worm gear box looked OK, but I added grease for good measure. One fiber washer on the armature spindle didn't match the three others, so I replaced it with one of the correct size since I had everything apart anyway.
Ever since I've had it, the fan has behaved like the blades are slightly out of balance, and a bit of a jitter can be felt on the back of the fan head. I checked using a couple of basic techniques and I think the blades are about as trued up as they can be. I've had decent luck using trial and error in adjusting blades on other fans, including my 12" GE Vortalex, which has blades that are notoriously difficult to balance. If there's a sure-fire way of checking for proper balance, I'm all ears.
I think the source of the rattle is the oscillation linkage. If I put pressure on both ends of the linkage, the rattling is minimized, but I don't think there's enough clearance to add another washer at either end. In a quiet room the rattle is noticeable. With two people conversing or if TV is on or music is playing, not so much. Except for this rattle, the motor and blades make almost no noise and the oscillation is smooth as silk.
I'm running the fan now so its innards can resettle after having been messed with. The rattle is at least only intermittent now. The poor thing was running for more than 12 hours at a time during our hot spell this past week, so it could be a component that's finally wearing out.
Should I consider that the fan is nearly 90 years old and leave it as is, or might there be something else I can do?