"There is scarcely any disease -- even AIDS or tetanus -- for which there are no instances of spontaneous recovery"
Rabies. There have been a few "recoveries" that appear to be legitimate, but they were hardly "spontaneous".
I haven't looked at rabies. the survival rate after onset of symptoms looks pretty grim, but perhaps there are people with natural immunity.
At any rate, even with modern medicine, most people recover or experience remissions from most illnesses without intervention. Quackery thrives in this arena.
The problem arises when quack medicine invades a space for which the disease is potentially fatal and the medicine is generally effective.