My belief as to why Sweden keeps the royals around is that Their Majesties provide a direct link to Sweden’s past. If one were to take into account that when you listen to the King speak about the goodness of the Swedish people, you sense that your father, grandfather, great grandfather and so on also listened to that kind of reinforcement.
It’s the idea that this King is related to your descendant’s King that is the comfort. He is a Swede as you are, and that generates the pride in being Swedish. The King (or Queen) is the living touchstone to your nation’s past.
As an American, with the disposable leaders we have, one gives deference to the Office of The President, not to the guy who warms the chair.
Group identity is destructive unless it works according to organic traits, or things we have naturally in common without having to be indoctrinated in them.
This is why one nation, one culture, one language, one set of values and one heritage is always a workable form of government, and our modern conception is so offended by it — we fear it because it’s what will replace our cosmopolitan, consumerist, mass control societies.