Passenger Pigeons were regarded as a menace by early settlers, like locust. And like locust, they were eliminated.
To go from 136 million in 1871 to zero in 1900 (the year the last passenger pigeon was shot in the wild) would have taken a phenomenal killing effort. At that size of population the reproduction rate must have been getting on for 100 million new birds a year, and every bird killed must simply created a better chance that next year's young would survive, because they would be competing for food with a smaller flock.
Granted, the nesting areas were relatively small and therefore subject to easy destruction, but two (related) factors should also be taken into account: disease and invasive species (which could well have brought diseases with them.)
Although introduced to late to be the culprit with respect to passenger pigeons, the common starling is an example of the massive effect invasive species can have on local ecologies. Furthermore, the massive changes to the prairie eco-system as the result of farming must have had an effect as well.
So while hunting and wanton destruction of nesting habitat obviously didn't help, it's interesting to ask, "Could the passenger pigeon have survived even without deliberate attempts to kill it?" The answer is not obviously "yes" (nor is it obviously "no", which is why the question is interesting.)
In this context it is worth remembering that the exclusion zone around the worst civil nuclear disaster in human history is far, far better for the local wildlife that simply having a thriving human population in the area: http://www.slate.com/articles/... (the article incorrectly states that observations of wildlife diversity around Chernobyl depend on the assumption that radiation isn't as bad for animals as humans, but this has causality backward: it is simply a matter of empirical fact, backed up by systematic observations carefully ignored by critics, that wildlife diversity in the exclusion zone is as high as that in protected nature reserve.)