Nice counter-quote from Shakespeare. I might want to write something about that, so thanks for bringing it to my attention.
But I think that dispute turns into a semantic one. The dichotomy I draw is basic to human experience, so there's no sensible disagreement with it. The disagreement is about the labels and analysis of relevant concepts. What counts as "natural"? I've written a lot about this, and the humanist distinction between civility and wildness is a centerpiece of my philosophical writing.
Certainly, nature makes us, but precisely because nature is wild, nature outdoes itself in doing so. Nature isn't wholly unified, but is a multitude containing contradictions and incommensurable varieties (such as the trillions of planets, or the divides between the subatomic and the macrocosmic, or between black holes and spacetime).
If you think we're wholly natural beings, would you countenance wild behaviour from your neighbours? Or do you see no difference between humans acting like wild animals, and humans like civilized people?