Boring because it's the standard approach a lot of kingdoms use; you're already advocating for not having a spiritual caste because it's rarer. It's unprofitable because the vast majority of current polities don't do a monarchy, or do a nominal one, in the case of Britain, Japan, and some of the northern ones. I'll admit that being trained to be a leader is an okay method if it goes right, but a) a lack of decent parents b) genetic issues c) estranging power from the people and d) making people just assume they deserve the position has historically led to markedly incompetent rulers. In other words, it trades Adaptability for an extremely iffy Competence and Stability, or possibly even Adaptability and Competence for Predictability, in the sense that everyone can thus focus on a family, who may or may not actually be suitable.
Additionally, a monarchy goes against our commune-leaning ways and our history of overly prideful leaders. Most monarchies based their rule on a claim to a) all the land within their territory - which explicitly goes against our "everyone works the land and gets a share" - or b) divine right, which goes against our lack of religion. These monarchies also usually fell once currency became detached from land ownership. I.e., when the economy shifted away from a rentee setup where nobles fought with each other over how much land they could have others work for them. You maybe aren't looking for a noble class, but having a monarchy/special family would most likely lead to one regardless, as families track their relation to the monarchy, and hierarch themselves accordingly. Which results in the main family considering themselves as deserving of this position, and whatever else they want.
"Greek style" democracy - or any of the number of other methods of electing a leader - mean that you elect someone based on merit, selecting from a much wider pool of candidates. Yes, the resources used to train the candidates would be spread out, but any family or person still aiming for the position would be training themselves appropriately regardless. (With notable benefits as the education required becomes more widespread, tutoring methods become better refined through practice and competition, etc.) There furthermore would be higher chances to find a notably excellent person, who has the higher drive necessary to rise to the occasion. And, finally, our Big Man-in-waiting process would work to ensure that they learn what they need to be a good leader.
Also, do we even use greek style democracy? What the hell are we?
