Check out this blog post: 3 reasons why New Zealand has the best-designed government in the world.
And here’s the topic relevant quote:
Monarchs are more effective than presidents precisely because they lack any semblance of legitimacy. It would be offensive for Queen Elizabeth or her representatives in Canada, New Zealand, etc. to meddle in domestic politics. Indeed, when the Governor-General of Australia did so in 1975 it set off a constitutional crisis that made it clear such behavior would not be tolerated. But figurehead presidents have some degree of democratic legitimacy, and are typically former politicians. That enables a greater rate of shenanigans — like when Italian president Giorgio Napolitano schemed, successfully, to remove Silvio Berlusconi as prime minister due at least in part to German chancellor Angela Merkel’s entreaties to do so.
Napolitano is the rule, rather than the exception. Oxford political scientists Petra Schleiter and Edward Morgan-Jones have found that presidents, whether elected indirectly by parliament or directly by the people, are likelier to allow governments to change without new elections than monarchs are. In other words, they’re likelier to change the government without any democratic input at all.
This is a seriously strange result, but perhaps it’s not too surprising. After all, representative democracy can be seriously messed up. If anything, what this post shows up that we would need to move towards direct democracy to have a system that is truly superior to constitutional monarchy – because the current representative democracy is an empirical democracy fail!
On the other hand, a just-for-fun monarch would be cute – perhaps as state mascot ;D