Mice. The issue was due to mice.
They'd packed the clutch so full of debris and droppings that the disk and fingers could no longer press together. I'd had the tractor taken to the local dealer on a flatbed, and they were the ones to discover the issue after they split the tractor. I didn't get to see it myself, but they said they couldn't believe how tightly everything was packed in, and glazed from the rubbing. Some parts were repairable, others were replaced. 9.5 hours in labor and $760 in parts and part repair, for a total cost of $1800. I was... upset.
There was a drought here in southern NH, and apparently the mice were invading vehicles and homes very aggressively in search of water. Mice in another garage damaged my wife's motorcycle, and ate a wiring harness in her car that's parked out in the open during the summer and was driven regularly. I didn't have any mouse problem at all my first 2 years on this property. One garage can be made mouse-resistant fairly easily, but the one that houses the tractor needs a lot more work. I'm trapping for now; about 100 over the past 2 months.