Federalism
Before we talk about anything else, let's note that the United States is a federal system. Each state sets its own rules. Even the titles might be different. So the following discussion will be centered on how things are done in Pennsylvania, which may be quite different from how they might be done in other states. Another state might even work more like you expect.
Judges
It's worth noting that in Pennsylvania, the common pleas, commonwealth, superior, and supreme court judges are only elected when they first take the office. After that, they stand for retention every ten years. Only the magistrates (the lowest level of judge) stand for reelection.
In terms of stability, judges almost never fail retention. For example, Russell M. Nigro is the only Pennsylvania Supreme Court justice to have lost a retention vote since they were established in 1968.
School directors
I'm not sure that it is clear what a school director is. In the US, there are typically five levels of government: municipal; school district; county; state; federal/national. The municipality could be a city but states have various other options, which may include town, village, borough, or township. The municipal selection will vary by state. The relationship between municipalities, school districts, and counties is not clearly hierarchical. For example, there are multiple counties inside the municipality of New York City, which I believe has one school district.
A school district may contain one or more than one municipality. Or it may have the same boundaries as a municipality or county. Anyway, the point is that the school directors are the council for the school district. They set property taxes and approve the budget. They typically meet once a month for a stipend rather than a salary. They hire a chief executive of the school, who is a salaried employee. They have the same type of qualifications as any other council position. Educators may have a step up the way that lawyers do in more legislative positions.
Unlike your other examples, school directors are not full-time employees. They are elected representatives like county commissioners or city council members.
Pros and cons
Some might argue that voters have as much ability to evaluate judges and coroners as elected representatives do. On one side, this is true. The special quality needed to be elected is popularity, not judgment. On the other hand, most voters don't actually do this. They vote for the candidate endorsed by the local party. So the result is essentially the same as if the local executive chose them; the party rewards those who support it.
It is possible for voters to choose a different person from the party selection. It just rarely happens. When it does happen, they don't necessarily choose better.
For regions that are relatively balanced in partisan terms, having offices be elected instead of appointed by the government can have an auditing effect. A treasurer or coroner of the opposite party from the executive or sheriff can review certain cases and ensure that money is spent as it should or that justice is done. But too often they are of the same party and thus reluctant to actively disagree with each other.
The real weakness is that if the voters choose, each individual voter has little reason to take the time to understand the qualifications. So they vote based on other estimates, e.g. party or newspaper endorsement. Elected politicians at least want to avoid later scandals. So they don't want to appoint someone who will later prove obviously unqualified. They are more likely to pick milquetoast candidates than unqualified ones.
Of course, the candidates chosen by politicians won't be the most qualified. First, the politicians don't know who is most qualified. They know who has supported them. They tend to pick strong supporters with moderate qualifications. Just as they do in picking who to endorse in elections where voters choose. Neither politicians nor voters are really set up to make the best picks.
If we really wanted to maximize qualifications, we'd come up with a system that worked more like a jury. Pick a hundred voters randomly, pay them a stipend, and let them choose the candidate. They can take the time to really evaluate the candidates, knowing that their vote was actually important, not just one of thousands or millions. The stipend should be enough that they can take time off from their regular job to do this one.
Or we could maximize stability by allowing judges to select their successors. Judges are far more qualified to tell if someone is qualified and has the right mindset. The incentives would be to pick mindset more than qualification, but not by so much that candidates would be much less qualified than sampled elections. They would probably be more qualified than those chosen by voters or politicians.
Political appointments maximize unity. The judges will have similar beliefs to those of the politicians that appointed them.
It's like with anything else. How much do you value control (direct election) over best qualifications (sampled election) or unity (political appointment) or stability (successor selection)?
