Not that it's my prediction, but evolution can also work by all humans dying due to the destruction of their environment and then something else evolving of whatever creatures survived. Human extinction is not different from dinosaur extinction; I believe that we'd rather avoid it for selfish reasons, regardless of the fact that it's just one paths of evolution among many. Certainly it being an evolutionary path not unlike others does not make it tempting.

"Evolution" is whatever is happening - survival of the survivors. If humans kill a certain number of badgers and the number of hedgehogs rises plus a bunch of other hard to predict things happens, it's "evolution" as much as humans having killed larger predators and done nothing about badgers and whatever happened next. An alien observer might comment that the number of badgers went down due to humans, or instead that the number of hedgehogs went down due to badgers. The only real question is whether you think more harm than good is done by reducing the number of badgers, and it ought to be answered on specific grounds.