I know it's difficult to think about such a complex process as a multi-species epidemic, but it happens. The best studied system involves human beings, pigs and birds, all living in China. They pass influenza back and forth, species to species, and every now and then create killer flu.
Yes, this is the perfect example which does not apply to the theory in question. Humans in China live in very densely populated areas, close together with their domesticated animals, and this fact allows them to pass diseases to each other. This does not apply to animals living in the wild, animals who also are too spread apart and thinly populated to quickly pass a disease, and thus create a contagion within their own species, assuming they are even capable of being infected by the cross-species disease in question. It's a very weak theory. And again, where is the species made extinct by a disease?