In other words, if the corruption is not just particular localities, but infects the national government as well, then we're all screwed?
They get stuck with it because 1st world countries insist on continuing to send aid even though the corrupt governments take all of it and it never benefits the people. Not to mention 1st world countries giving legitimacy to those corrupt governments in many other ways.
In other words, on a global level, the idea that corruption in 3rd world countries is "a global problem" has made the corruption worse, not better.
The US is a union of states. Although we have a pretty massive and powerful federal government, many things are the responsibility of the states and the federal government has limited ability to intervene. Using the FBI is one of the avenues I believe though.
Moreover corrupt cops thrive because of close knit relationships between local DAs, local law enforcement, local judges and local politicians. A national wrecking ball is needed to break these toxic corrupt relationships and a strong, national institution is needed to create the institutional inertia against systemic police corruption from arising again.
You can't vote corruption out of office. It has to be rooted out from outside. Even the cleanest politicians cannot operate from within a corrupt system.