It always seemed like a false choice to me. It's not like humans need to literally hunt a species to extinction to wipe it out. We can (and have) done it by disrupting the food chain, altering environments and introducing invasive species as well.
The push back against ancient humans causing or at least, significantly contributing to extinctions feels odd. Few would argue that human activity, even absent hunting or advanced technology, can threaten species today, but for some reason it's considered impossible for ancient peoples.