Well, humans moving into ecosystems have been driving species extinct presumably since we developed civilization. Thousands of years maybe? Climate change has been affecting species extinction over what time frame? 50 years? They are tangentially-related but not inextricably linked to our effect on other species. Let's just say it's (climate change) a factor currently, among the many other factors that we affect that can drive species extinct. Such as feral housecats which have driven dozens of bird species extinct over the last 100 years alone, and that problem is almost entirely anthropogenic in nature.
Humans are destructive by nature. Agent Smith wasn't far off in his assessment of humans in the Matrix. We consume and destroy. I don't think it's very arguable that the planet as a whole would be better off with far fewer humans, and even better with none at all.
For the record, mother nature has driven far more species extinct than we ever will, unless we get to a nuclear winter scenario.
I just really doubt climate change is the nuclear winter.