Thanks for the reply.
Perhaps "bearing" isn't the right word. Where the shaft passes through the face of the fan and connects to the blades, there is a small hole, and it actually appeared to have a cloth ring around the hole. I lubricated the shaft and put oil on this part too since this seemed to be a likely spot where metal could potentially rub metal.
There was grease in the back assembly that houses the oscillation mechanism. I cleaned out and replaced what I could from this part (it is assembled so that I cannot open it fully without damaging it). But I don't think this is the problem because I actually ran the fan without reattaching the oscillation unit, and I had the same thing happen - it ran for a few days, then died.
I have also pulled the entire rotor assembly out of the winding (I didn't need to mess with any of the winding to do this), and lubricated the face of that too. I noticed when it died again that this part was bone dry. I figured that maybe I didn't use enough oil, so I lubricated this part more heavily the second time, and the same result - it died a few days later and when I opened it, it was bone dry.
I haven't found any reservoir for lubricant, and I was wondering if perhaps I'm not using the correct oil. Perhaps the motor runs too hot and is burning it off. But I have visited forums that discuss antique fan restoration, and the 3-in-1 oil is what they say to use. I am hesitant to use a grease for fear of gumming up the rotor and damaging it.
It doesn't seem like the capacitor is the problem because I have already replaced it, and after lubrication, the fan will run fine for a few days before dying again.
I'm hoping that it's not a problem with the winding because that would likely just require me to replace the fan since it would be cost prohibitive to rewind it.
Thanks again for the reply.