No it's not evidence for your idea that animals 'heal themselves' by fasting. It is evidence for what other posters have already said, that animals who are ill stop eating and become less active.
I've owned many different pets, worked in wildlife sanctuaries and spent ten years in the hell that is the NHS.
In that time, whenever a person or animal stopped eating and we couldn't find a way to get nutrition in somehow, they generally went ahead and died.
Racking my brain, I can't think of more than one or two occasions when someone who had stopped taking food and fluids made any kind of recovery. It's happened, but it's rare. This is just anecdote of course.
Sorry I am re-posting as I got one idea.
You have obseved that in many cases animals died due to not eating but in few cases they also recovered. Can we take it like that those animals which were underfed or fed normally died but those overfed(obese etc.) before sickness recovered?