Representative republics are a perfect example. The most effective administrators and policymakers tend to be intellectually curious, well-versed in law and economics, good at taking feedback, at least somewhat good at thinking creatively, and morally upstanding. In principle, good looks, oratory eloquence, a charming personality, well-connectedness, and personal wealth are not particularly useful to creating and executing government policy. But the primary selection criterion for public office is the ability to win elections. Thus, that is the target, not being a good policymaker. And being charming is far more useful than being studious when it comes to attaining public authority.
Hereditary monarchy is of course quite prone to this problem as well. You cannot make the selection criteria opaque, because that would cause constant succession crises. So, it has to be something transparent, such as a hierarchical list of heirs which determines who has legitimacy as a new ruler and who doesn’t. Even if the rules of succession are immutable, they can still be gamed, by assassination for example. Therefore, factors such as getting away with poisoning your elder siblings or sending your nephews to battle on suicide missions become crucial in determining who gets authority.