There are many state-level offices in the US that are elected by the people other than the legislature and the Governor, in the executive (Treasurer, Controller, Attorney General, Secretary, oil & gas regulators, Land Commissioner, Insurance Commissioner etc) but also judges. It's the same at the county/city/district-level where several offices apart from the mayor/manager and the board are elected, like the judges, DA, Sheriff, school board, community college board, transit board, etc. I don't know of any county which allows such a large array of officers to be elected directly by the public.
I have a hypothesis that such a significant difference in power is likely to have some effect on governance and the concessions that people can extract if voting has any effect on government policy in general. [1] What does social/public choice theory have anything to say about its effects?
[1] Let $X = \{X_1, X_2, ... X_n\}$ be a set of choices of candidates for an office. Let $Y = \{\{Y_{1,1}, Y_{1,2}, ...Y_{1n}\}, ..., \{Y_{n,1}, Y_{n,2}, ...Y_{n,n}\}\}$ be a set of policies that each will enact if elected such that $X_n$ corresponds to a policy set $\{Y_{n,1}, Y_{n,2}, ...Y_{n,n}\}$. A rational voter will choose from $X$ whichever has the policy set with the most number of desirable policies (or the least number of undesirable policies). (Consider any voting system immune to Arrow's Impossibility Theorem, like score/range voting.) But if the powers of the candidate were divided into $n$ independently elected political offices and the number of candidates remained constant between the two cases (i.e. $n$) such that each candidate could enact only one policy, the voter could choose a candidate for each of the $n$ positions with the most desirable policy. While in reality, a candidate for an independent elected office may enact more than one policy if you define the policy set differently, it is clear that even in that case, the voter will be able to choose more favorable policies.
