Who does the selection? In the USA, doesn’t it always come down to voter choice one way or another? We typically elect a sheriff but a mayor appoints a police chief. The mayor is elected however. What process could be used to determine process and selection that would avoid voter input?

The highest level public offices should always be either directly elected, or appointed by an elected person for only the duration of their office. This way voters can get rid of them if necessary. Otherwise they become an unelected deep state.

Other levels of public office should be strictly by competence. Meaning by trials open for all citizens.