Well the monarchies of real life held most of their authority because they were related to higher powers that people thought gave them authority.
The Europeans had The Pope to declare kings had god's permission to rule.
Caliphates claim relation to their prophet for legitimacy.
Eastern Empires expressed The Mandate of Heaven over their people.
Even Pharaohs claimed to be direct descendants from gods.
The butterflies don't have any of this though, they literally just have the power to shoot people who say they don't rule with magic.
I believe that this has so far given them the right to rule because people believe that only their bloodline has the power to wield the wand, sort of like an excalibur situation.
I have a feeling that if the common peasants found out that the wand works for any asshole, including the nobodies that the current Butterfly line is made of, or a literal human from earth, the entire Monarchy would fall in favor of a true dictatorship based on fear and power.