My take on it, personally, is that the primary benefit of a monarch is precisely that they are inherently illegitimate, in a way. This means that every day they have to justify to the people and keep answering the question "Why do we keep these people around?"
A president only has to convince (fool?) the people twice, and can always hide behind a "mandate" from the people to justify bad behavior. A monarch has to fight to earn that legitimacy every day. Plus they have the incentive of not screwing their kids' inheritance up. That's the value I see in it anyways, I don't know if there's a more sophisticated articulation of this idea out there.