There is always a trade off when we decide how to select our executives. We have to make a choice between democracy and efficiency.
It would certainly be more efficient (most likely) to appoint county executives. It would be better because we would be more likely to get executives who are actually good at their jobs. When we elect someone as a county executive, we are typically electing them based on their ability to campaign. The ability to campaign is not the same as the ability to manage a county government. Thus, appointing an executive would be a surer way to get someone who was actually capable of running the government.
However, appointing an executive is less democratic. We do not want to live in a country where people govern us because someone has decided that they are best qualified to do so. Instead, we want to select our own leaders. When we appoint executives, we are losing our control over them. This is more likely to lead to a situation where the county government does not really do what we want it to do.