A lot of our ancestors got by by using the 'big-black-stew-pot'. Anything and everything that might be eaten was skinned, cleaned, and dropped into the pot. The pot was left at a low simmer during the day and covered and allowed to cool at night. In winter, when you have a fire anyway, it might cook slowly 24/7. Over a banked fire at night.

In good times it was small animals like squirrels and rabbit and the occasional cut of beef or venison. In normal hard times it included any small animals. Mice, small birds, lizards. When things got desperate bones and skins went in. Bones were saved and reused until they disintegrated.

I suspect that even a rotting, and diseased rabbit would be safe to eat if cooked long enough.

If you insist on eating your meat rare or letting the cut do little more than 'kiss-the-pan' I would think your going to have to be more selective. Stewed for a day is safer.