The current FBI exists for one purpose, and it isn’t the police functions you noted: it exists to disrupt and degrade any organized opposition to the self defined Federal State.
Basically it’s the guardian of the new aristocracy - the bureaucrats and a few politicians who are seen as loyal to the bureaucracy.
The German nobility had their own hired guards who did this - harassed and intimidated any peasants who tried to organize against them. Of course so did a lot of tyrannocracies - think of the Varangian guards in Byzantium, or of course the People’s Commissariat for Internal Affairs
(NKVD). They all do it. They all know you have to do it if you want to successfully dominate people.
That’s the only real reason for its existence.
In other words, America's KGB