Well, you have to realize that the one species that is threatened is often threatened because of other reasons. When you have that, on top of having predators that are maybe going hungry, then yes, they can drive it down. In academics, we talk about an extinction vortex. Once a species is driven down to very low numbers, then all of these factors begin to push them into this extinction vortex. When we arrive at that stage, absolutely, we have to become very proactive.
However, we must never assume that in nature we have very linear interactions. If we look at the food web in the Atlantic ocean, it is phenomenally complex. Yet there are cod and seals, and seals do eat cod and other fish. However, it's not necessarily so simple that we can say that by reducing seals we will increase cod, because of this highly complex, interactive, connected system we work with. Taking this holistic approach and looking at the whole ecosystem I think is what is needed.
Sure, when human life is in danger, then that probably requires a special case. With the coyotes in Cape Breton, those coyotes need to learn that humans are not food--they provide food--and humans need to learn they should not provide food.