Thankfully Google quickly pointed me at lots of sites that told me the spider will corrode out in time, and I found the extremely helpful videos provided by Appliance Parts Pros. With the video guidance, I took things apart first to confirm exactly what was broken, and sure enough the spindle had completely separated from the spider, hence no rotation! It turned out that the rotating drum had obviously been rubbing on the plastic drum for some time, because when we separate the two halves of the plastic drum the front had rubbed all the way through. So we needed a new front half of the drum as well. Just as well those are available. I suspect next time this happens (7 or 8 years I suppose) none of the parts will be available.
Overall it took about 2 hours to take things apart, and about the same to put to back together again. Nothing too difficult; just fiddly.