I think part of the point of the thought experiment is that.....the people who would set up a society like this think that there ARE occasionally times when it might be a good idea to launch a nuke, so they are making sure that a leader would only launch it in times of direst need, when they can manage to justify to themselves that killing a child really is worth it in order to launch the missile. And nuclear disarmament would not allow for that possibility.

But that approach still doesn't work for me, which is why I'm on your side of the whole thing not making sense and being a bad idea.

The story doesn't spend a lot of time focusing on your concerns, because it's from the perspective of the child and she has other priorities in mind than ruminating on the history of the political system that set things up so that she's in the position she's in. I don't know if you'd like the story or not.