Sears Catalog number is 57C 2160 as seen in the 1975 Sears Wishbook. It appears to be a model 564.21600300
It sounds like the capstan and pinch roller near the head is working fine, so it plays.
My suspicion given that the unit is older, is either the drive clutch on the takeup spindle (if it has one it's right under the takeup spindle) could have worn out or is missing a part like the spring that puts pressure on the clutch. The cleaning cassette that you ran is short, and since these units would not have autostop features (to my best guess) the takeup drive could keep spinning while the spindle can't turn. This could ruin the clutch or possibly break teeth on a gear from shock since they would likely be plastic and old. This would explain why the take up reel isn't turning, while the capstan and pinch roller do. I believe the capstan is belt driven, and since it works the belt is likely fine.
I'm not sure how to check these parts except by disassembling the unit. If you disassemble it you will be able to see how it is driven. They may have used belts and the belt was allowed to slip deliberately to function like a clutch since this was an inexpensive model made in Korea at the time.