Add in the fact that they literally were trying to kill him already, hence Theon having to protect him. If it needed to be the Night King killing him, they could've just even had Bran say something like "he needs to kill me personally, because, uh, let's go with...it's been foretold in some gay prophecy, maybe? That'll do." Fucking just something, anything.
It's weird how so many dorks' only problem is Arya killing the zombie king, when that in itself is perfectly legitimate based upon what's been established - that she is the ultimate assassin. Silly, but explainable.They just could have had an actual reason for the Night King to put himself in danger.