Without the h3 pawn, white could capture at g2 after black moves from f3 to f2. Edit: Regarding your second point of getting a mate in one, I think you are mis

Without the h3 pawn, white could capture at g2 after black moves from f3 to f2. Edit: Regarding your second point of getting a mate in one, I think you are misunderstanding the goal here. It's to construct a mating sequence that is as long as possible, and which permits no shortcuts (in this case the position has a mate in 2 but no mate in 1). A shorter sequence can always be constructed by taking the end of a longer one, so that's uninteresting.