If one must have a certain mind to see things in a certain way, you can be sure you're dealing with a healthy dose of perspectivism. And that's really the central argument of Wallace Stevens's "The Snow Man." In Stevens's world, the reality outside of the realities our individual imaginations create is, well, boring, and that very idea is the hinge on which this poem swings.
This poem has one big hole in its logic: snow is snow, no matter what kind of mind you've got.
This poem isn't perspectivist at all. It's nihilist, and in the end, Stevens proves that nothing exists. At all.
The natural world is just about the perfect setting for "The Snow Man," and that's not because snow men just look so much better against a backdrop of leafless birches and evergreen pines. It's because the natural world is supposed to be outside of the human world. It's not something we can change or control. It's Mother Nature. But in this poem, Stevens turns that notion on its head, and proves to us that the natural world is, like just about everything else, all a matter of perspective. We may think the world turns and the seasons change and there's nothing we can do about it, but Stevens begs to differ.
The natural world in this poem is entirely irrelevant. It's just a tool Stevens uses to talk about ideas.
The natural world is essential to this poem because Stevens is talking about the kind of philosophical experience you can only have in nature.