Over the past decade in California, a number of ballot initiatives have gone to the voters in various municipal and county elections to eliminate elected offices by consolidation.  The most recent that comes to mind is the County of San Mateo proposal a few years ago to combine Sherrif and Coroner.  Voters on the Left Coast seem to be rejecting these ideas uniformly as they like the opportunity to elect their officials.  Budget impact seems like it would be minimal either way from analyses I've read. 

School boards are another great example.  Despite having a State-elected Education Secretary, we continue to have multiple, overlapping school districts in every geographic area.  Imagine the cost savings of thousands of elected school officials giving up their jobs so  that the State elected official could run things, yet nobody seems to like that idea.