I found this to be an interesting remark. Living in a Constitutional monarchy myself, maybe I should try to clarify.
Actually there is not a significant difference between our constitutional monarchy and your republican monarchy. The basic difference is that head-of-state is also the executional head-of-state and politically elected. Here, our head-of-state is mostly ceremonial and used for PR. The prime-minister is the executional head-of-state, meaning that he is the one actively leading the government. Like your president, he is elected by the people. The King has no saying in the daily governing of the country, however he is sometimes used as an impartial mediator, being officially apolitical and above the parties.
This of course is the theory. But if you want to talk to the active head of the state, you talk to the prime-minister, not the king.