A few things to say about this...I voted yes, with the condition ... it should be used to reintroduce those species that were not wiped out by natural selection, but rather unnatural selection
One species wiping out another species, particularly by choice in only natural in the sense that we naturally evolved the means to destroy another species. This is clearly not the basis of evolution as a concept or as a whole, and it does not follow, in any way, the natural processes that historically cause the extinction of a species.A point already made by RA_Pierce, but how is extinction caused by humans NOT natural?
As to the food chain, it's typically the introduction of a new predatory/carnivorous species that throws off the balance.
Even with herbivore species though you risk causing problems if those herbivores, like the mammoth, consume too much of the resources available.