From my viewpoint, royalty is a kind of icon; and in the way icons relates those praying and the deity, royalty relates kingdom and subjects.
Human beings are mostly followers, so we need leaders but most of the time those leading cannot spend time with their followers, so the best way is to find a face to accomplish this and let the leader free to guide the rest.
I think that to define modern royalty we would say: The King or Queen, reigns but does not rule. A midway between Louis XIV, L'État, c'est moi, and Napoleon II.
A point in favor of royalty is, as an English friend told me: "The Queen represents the people, she integrates us as a nation." A notion easily replaced by sovereignty and the people themselves, but this is an abstract concept while the King or Queen is a flesh and bones reality and can be accepted readily.
The science of politics nowadays seems to move towards democracy, whatever this means, but royalty still keeps the glamour of the Royal Court and Aristocracy.
Besides, I must recognize that the Royal Wedding of Charles and Diane brought magnetism and excitement to people around the world, and profit to many, for the fairy tale ingredient that presented Diana as a modern Cinderella.
Is it more useful than other ruling systems? I do not think so.
It has glamour, it has a kind of corny attraction to crowds, it can be seen as the representation of people but it does not look much better in action when compared to a Republic or Federation.