Imagine you were starting a country from scratch, would you pick a monarchy as a system of government? Or would you pick one if you weren’t personally going to be the leader? Probably not. If you care about this sort of thing at all, then it is hard to see past some kind of representative democracy with an elected head of state. If you think that in an important sense all people are equal and should count equally, regardless of wealth or talents, because we all have the same capacity to love, to strive, to suffer no matter what our position in life, then the idea of raising a king or queen above everyone else, simply because of an accident of birth seems abhorrent.
Despite this, I have think you can make an argument for constitutional monarchies as a form of government. You could reasonably choose to create the first country on Mars as a monarchy, for example.
In practice, the best examples of monarchies are at least as good a place to live as the best examples of republics. Compare the UK, Japan, Denmark, Belgium, the Netherlands, for example with France, Germany, Ireland and the US. Surely the quality of life is comparable across the groups. Why is that? Well, you have to fit the monarchy into the rest of the system of government. The monarch is the head of state, but the business of government is delegated to a democratically elected parliament.
You can decouple the idea of monarchy from the idea of dictatorship. Dictatorships are bad whether the head of state is a king or a president (think of Mugabe’s Zimbabwe, Castro’s Cuba, or Franco’s Spain).
I would argue that the idea of the common humanity, which initially seems antithetical to having a monarchy, is actually a good fit. A royal family is a family going through the same ups and downs as any other, celebrating marriages and births and mourning deaths and set backs and so on. Because they are always there, the public sees them over the course of these events and can identify with them. A president governs only for a short term so their private lives are only visible for a span of time and the president can figure in a more abstract and ceremonial capacity in ordinary people’s lives.
Executive presidents, like in France and the US, are party political, which stops them representing everyone. The monarch doesn’t pretend to represent the population in theory or in practice. But they belong to the population in a certain way. They can’t really emigrate. They are tied to their country.
Is it fair that some people should be fabulously wealthy because of an accident of their birth and live in castles and palaces and spend their time in a perpetual state of leisure? Life isn’t fair. Wealth and talents are not distributed evenly and cannot be distributed evenly in any system of government. This is as true in republics as in monarchies. If anything disparities are more offensive in a republic where the myth exists that all people are equal. People compare themselves to their peers. The royal family are out of that sort of calculus because they have no equals. I wouldn’t want to trade places with royalty and don’t believe I know anyone else who would.