The Harbinger mission is different from the Hawkmoon exotic quest (Let Loose Thy Talons). I believe you're thinking of the final mission in that quest, "The Crow and the Hawk." That's the mission that unlocks the Hawkmoon. Harbinger allows you to farm for random rolls of the Hawkmoon. Your account doesn't show you completing, or even entering the Harbinger mission. You'll have to complete it once, and you'll then be able to launch it from the EDZ map.