Joseph, even in a constitutional monarchy, if it’s a real monarchy the king is the master, and the people are his subjects. People swear loyalty to the king, and attempting to kill him is treason. He is bound by rules, but that’s nothing new; even the kings of Beis Dovid were bound by rules.
The elected government governs on his behalf, so that he can stay out of politics; but they work for him, not for the people. “L’etat c’est moi” is not just about Louis XIV; it is a plain factual statement of how a monarchy is organized. Just as in the USA the “executive branch” is the president, in a monarchy the “state” is the king.
(In the UK, since about 1700, there has been a separation between the king and the Crown; but the Crown is not a personification of the people, it’s a personification of the monarchy. It’s a sort of abstraction of the king’s “kingness”, as opposed to his personal attributes. The Crown is the king, just not him personally.)
The founders of the USA did something very different, and made the president a servant of the people, not the other way around. We are not his subjects, he is ours. Just look at how criminal cases are brought. In the UK or Canada, it’s “R v Smith”. In the USA it’s not “The President v Smith” but “The People v Smith”.
This makes it appropriate for a king’s subjects to pray for him and honor him, whereas doing so to the president is just the same as doing it to a janitor. He’s just an employee doing a job.