You've never seen a live dinosaur, right? Or a saber toothed cat, or a giant sloth, or a mastodon, or an actual Neanderthal (some current individuals do come close) either. All of these creatures have gone extinct, most with no help from humans. Yet the earth, and life, persist and flourish.
What's my point? Only that critters going extinct is a natural part of the existence of life on this planet. The fossil record shows species rise, flourish, wane and die. Life goes on. Environmentalists bewail this very natural process.
Many species coexist with us easily, like the robin building her nest on the beam of my back porch or the mule deer chewing cud in the backyard. I agree we shouldn't go out of our way to eliminate most species. I'd make an exception for the mosquito which preys on us and worse, transmits serious disease, giving us the near-duty to eliminate it.
On the modest acreage of my winter place, rattlesnakes are somewhat common. My attitude is, if they stay away from the house and garage, they're welcome to all the gophers and mice they can eat. If they show up by the house, I kill them. Individual snakes are endangered, the species not so much. Over time those rattlers whose natural instinct is to avoid human habitation will thrive, those which lack this bias will thrive less.
Notice how nobody bewails the extinction of the smallpox virus or near-extinction of polio? Would anybody miss ebola if we could eliminate it? How about malaria, cholera, typhoid, dengue or plague? I, for one, could do without the lot plus influenza and not lose a minute's sleep.
There are things to worry about; the extinction of at-risk fringe species - large, small and microscopic - may not be one of them.