This very first sentence is incorrect. Zorn's Lemma does NOT say "for every linear ordered set every chain has a maximal element." It says "IF a linearly ordered set has that property, then..."

Your "counterexample" isn't a counterexample at all. The naturals do not satisfy the hypothesis of Zorn's Lemma. So Zorn's Lemma says NOTHING about the naturals with its usual order.