I’m not advocating in favor of monarchies here, but rather wondering how Bitcoin might affect/shift the balances of power in existing political systems and societies.

Is the Great War what Americans call world war 1? If so, those were not monarchies. They had monarchs, but those figure heads don't really deserve the name. Those warring nation states all had legislatures, which is where the real power was. The problem was the legislative bodies, not the monarchs, who were sometimes literally kidnapped and coerced into approving the desires of the oligarchs in the diets.

I don't think bitcoin favors monarchies. I think it extends their decision making process to more people, which is a boon to any system.