They had the sense to recognize that slaying the Night King would take out all of his minions. I believe they got that bit of knowledge from the expedition to the North to capture a zombie and bring it to King's Landing. In the initial battle where they ended up nabbing one of the undead, a white walker was killed and all those he had turned died.
So, while killing a 'lieutenant' won't kill the army it will certainly take out scores of zombies. Why wasn't that talked about? You've got a half dozen Valyerian steel swords that can take out an entire platoon if you kill the white walker in charge. Shouldn't Brianne and Jamie and Arya and Mormount and Jon be tasked with that job??