Yeah that is in no way a common occurence. If the injury itself won't kill the animal, a predator will. Seeing an animal in the wild that has recovered from a serious injury is very rare.

Of course other animals have immune systems too, so they can fend off a minor infection without any medical intervention, as can we.

But oh boy, if those bacteria and pathogens enter the bloodstream? Without medical care that animal is a gonner, be it Human or any other animal.