Lot of derision but not much explanation as to how this strange system came to be.

Many state constitutions in the US were written in the 18th and 19th centuries. The key differences relevant here to this discussion were that during these times, suffrage was far more restricted, and communities were far more sparsely populated and isolated.

Prior to the 20th century, suffrage was not universal and generally was restricted to wealthy white men of status, who, as a consequence of their socioeconomic standing, also tended to be more educated and thus better suited to rationally judge the qualifications of office-seekers. A consequence of universal suffrage is that the education level of the average voter goes down.

Most Europeans severely underestimate how few people lived in these states and across how much land they occupied. The US typically granted statehood to its territories when they reached the mid to high five digits in population. The majority of Western states are the same size as the largest European countries. Let me use California as an illustrative example. Its statehood was granted in 1850 and it had a population of 92,597. So just imagine essentially a group of people fewer in number than a single small European city trying to run a piece of territory the size of Germany (California is actually bigger than Germany by 69,000 km²).

What happens in such a scenario is that communities become very isolated and insular. They get used to running their own affairs since basically any model of centralised government is going to fail when your population density is 0.2 people per km².

Understand that aside from tightly-knit indigenous communities (who were branded as "savages" and categorically excluded from participation in so-called "civilised" society) this was literally unsettled land. Empty plains, dry desert, and wild forest for hundreds of kilometres around where there was no law but those of physics.

In these isolated communities, you still need to fill the required leadership roles, but you run into the issue where nobody is particularly qualified to these offices and further still, the townsfolk don't really want to just elect a single person to fill all the other offices by appointments. Rather the best way to fill these offices is by election where the community can get together and decide collectively who is best qualified for office. So how it would go is that everyone entitled to suffrage would, every other year, ride their horses into the county seat, which could take hours, and then listen to the candidates' campaign pitches, vote for whomever they thought was the most qualified for sheriff and county judge, and then go home and never hear from those people again for months on end.

As a result, when these territories were granted statehood, most delegates to the conventions that wrote the state constitutions saw no reason to deviate from these established methods for picking local office-holders.

Edit: I realise this also doesn't explain why these constitutions haven't been amended to allow for appointed judges in the modern US. The reason is because politics in the US is extremely cutthroat and anyone who proposes such an amendment is taking a rather unnecessary risk with their political career because their political opponents can then attack them for taking away power from the voters in favour of "unelected bureaucrats".