The advantage of a parliamentary/constitutional monarchy is that you get the head of state to get all the ceremonial stuff that people like, while the elected politicians have something from letting them get delusions of grandeur.

You see the problem in the US: they might elect a president, but when you look at all the ceremonial stuff, you'd be forgiven for thinking they're just an elective monarchy. "No one sits when the President stands up" and all that other nonsense: palace guards, pomp and ceremony, theme music. A few years ago, Canada's former Prime Minister Harper, just after he first won, tried to put that rule in place and his own staff told him to get stuffed. If he wanted to be treated like royalty, he could get himself into the vice-regal (and pretty much powerless) position of Governor-General and enjoy the trooping of the guards and the fancy salutes and all that jazz. Mere Prime Ministers aren't eligible for that cool stuff, they're supposed to work for a living.