Now everyone would not trust the mayor and the sheriff. And there would be gobs and gobs of tweets about "who was sleeping with who and that's why this kid got the contract to fix this road over here" and so on. With that entire influence campaign probably funded by the oligarchs from the wealthy city-state 300 miles away who want to use an area of your city for waste dumping or something.

There's no quick easy way out. We really do need to actually fix the problem. Restructuring the corruption won't help here. Mostly because it's still corruption.

Secondly, the idea that professional influence campaigns won't work because, "the target 'knows' everybody" is naive in the extreme. It betrays a lack of knowledge about what it is you're up against. These campaigns work, and they work well. That's the reason everyone from corporations to intelligence agencies put so much resource into them.

Lastly, none of what you said actually makes the mayor and the sheriff "not corrupt". Which is the fundamental issue. How do you ensure non-corrupt officials? Or at a minimum, decrease corruption. Your idea probably doesn't even shift the corruption, because corruption is already there. It just gives the corrupt local officials more money to steal in the end. Them knowing everyone increases the opportunity and possibilities for corruption. It doesn't decrease them, which is what you want.

What we need are ideas on how to be more effective in decreasing corruption. So less, "shift it to the local level" and more "reignite and reallocate more human resources to the public corruption segment of the FBI". But people will not want to do the latter, because it would catch their corruption. Whereas they would want to do the former, because it gives them more opportunities to participate in corruption.