Reading Raemon's recent Shortform on why the long-term leadership succession of small groups often deteriorates made me consider organizations which manage to successfully have several generations of competent leadership.
Reading through history, hereditary monarchies always seem to fall into a similar problem to what Raemon describes, with incompetent and (physically and mentally) weak monarchs being placed at the head of a nation, leading to a lot of problems. Republics, in contrast, almost always have competent leaders .
This makes life much better for the people in the republic, and may be in part responsible for the recent proliferation of republics (though it does raise the question of why that hasn't happened sooner. Maybe the robust safeguards implemented by the Founding Fathers of the USA in their constitution were a sufficiently non-obvious and important social technology required for republics to be viable on the world stage ).
A key difference between monarchies and republics is that each successive generation of leadership in a republic must win an intense competition to secure their position, unlike the heirs of a monarchy . Not only this, but the competitions are usually held quite often (for example, every 4 years in Denmark, every 5 years in the UK ), which keeps the ⸤fact that the office is competitive⸥ in the public mind very frequently, making it hard for the office to become a de facto hereditary position (a problem that happens with all too high frequency). By holding a competition to fill the office, one ensures that, even if the leaders don't share the same vision as the original founder, they still have to be very competent to be appointed to the position of leadership .
 One might disagree with goals of the leaders of republics, and they are too often appointed after their prime, when their health is declining [1a], but the leaders of republics are almost always very competent people.
(1a) - This (The fact that the leaders of republics are often elected when their health is in decline) makes me think it may be a good idea to have a constitutional maximum age, after which individuals cannot be elected to certain important offices, to ensure that only people who are in their prime (and hence likely sufficiently healthy) can lead the nation.
 - The existence of elective monarchies also is suggestive that the theory may be meaningful, but it again raises the question of why elective monarchies weren't more prominent. Maybe in practice elective monarchies were too likely to become effectively hereditary monarchies in all but name (c.f. the Hungarian kingdom and the Holy Roman Empire), that they didn't distinguish themselves enough to have a competitive advantage.
 - I contend that the usual way of appointing successors to small organizations (appointment by the previous leader) and to corporations (elected, but by a small body in a usually non-competitive fashion that is more similar to being appointed on a personal basis) is insufficiently competitive, and so is more similar to a hereditary monarchy than a republic.
 - Other frequencies for elections in successful republics include every 4 years in America (for the president), every 3 years in New Zealand, and every 5 years in South Korea.
 - One might look at, for example, the Congress of the United States, and question if they really are competent. Now, there is a good bit of competence in that body, but it does indeed leave something to be desired. This is a symptom of the fact that the US uses first-past-the-post (plurality voting) to elect its representatives, which leads to a two-party system, which is inherently substantially less competitive than a multi-party system, such as is encouraged by (mixed-member) proportional representation and approval voting. This happens for the same reason why a duopoly (an industry with only two major players) results in prices almost as high as are seen in a monopoly - you don't need to be good to win in a duopoly, you only need to be better than the other guy.