As one who uses a wheelbarrow nearly everyday, cleaning up after a horse, "so much does depend" upon a wheelbarrow. This may be a waste disposal question....after all, the red wheelbarrow is beside the white chickens. Perhaps, then, it is an existential question since all life involves those necessities implied.
Sorry, I could not resist the parodic response. However, I have often wondered if Williams is not himself toying with the proclivity of people to find greater meanings, and existential ones at that, in the mundane and banal--something like the movie starring the inimitable Peter Sellers, Being There, whose screenplay was adapted from the novella written by Jerry Kosinski. [Sellers's character, Chance the gardener, is a mentally challenged man whom a multimillionaire and president mistake for a brillant man who quotes Voltaire's passage about one's tending one's figurative garden when he is only speaking of his garden in back of the house in which he dwells in the beginning of the film.]
One thing I enjoy about this poem: Watching for the students to raise their heads from reading it with, to put it mildly, a myriad of expressions.