I think the possible outcomes of climate change (especially if we continue on our current trend) are unthinkably catastrophic, at worst including the potential deaths of a significant percentage of humanity. But there's still an enormous gap between that and "total extinction of large mammals". I don't think that's anywhere near a realistic consequence of climate change.

There is no conceivable future in which white-tailed deer become extinct in the next few centuries. We will probably lose some of the less resilient species, though.