Neutral in that it doesn’t habitually hide weapons and explosives (or hostages) on behalf of terrorist organisations. Neutral in that its (recent) doctrine is not to prosecute wars or support violence.
The UN has never been truly politically neutral either, nor could it, given the way it’s constituted. But, like the priests, their doctrine doesn’t include local staff actively assisting acts of violence.
I’m happy with the analogy, but thanks so much for your critique.