Humans really like having a figurehead, a living symbol of their country. A monarch provides this living symbol without giving them any apparent democratic legitimacy - it's obvious that the King wasn't chosen by the people, he's just luck of the draw and so if he tried to seize power that's obviously illegitimate.

In contrast, an elected President has democratic legitimacy and we see over, and over, and over what happens is that a President takes the opportunity to try to seize power. It used to happen that I'd explain this and then Americans, who apparently don't study history, would object that obviously this would never happen with their President ...