So many questions...
In part it deals with the concept of "federalism" whereas the national government holds significant power, but the smaller political subdivisions also hold significant power.
Federal laws are enforced by the designated federal agencies. We do not have a pure national police force, but we have the FBI, ATF, United States Marshals, The Secret Service etc. all with their niche of law enforcement activity as designated by Congress or the Executive Branch.
Now we have fifty states in the USA.
The leading "law enforcement official" in each state is called the "Attorney General" and YES he/she is a politician running for office every two or four years.
States have their own police forces.
States may set-up various State Authorities or Agencies that may have police powers especially in regards to specific crimes (e.g., arson investigations, probation officers).
States are divided into county governments, which may be strong or weak in relation to the parent state government.
Then we have cities and town governments, with their own police departments.
IT IS CONFUSING.
Maybe it is a case of diversity is a good thing.
There is no standard logic of how each state set-up their police forces.
Population density is a factor.
Geography is another factor.
What is the tax base to support the local LEO?
History, tradition and previous political turf battles have an impact - perhaps a separate large city "metropolitan police force" exists today because one hundred years ago a powerful politician wanted to create his own patronage fiefdom.
Also, I hate to break this to you Neil, but the interaction of law enforcement agents portrayed on American TV supports the writers' contrived story lines. Not how it is done in real life. Every police force cop and their commanders will fight for their designated piece of the law enforcement pie, eh maybe I should call it "donut delineation."
John McPartland
Well, but you can't expect to wield supreme executive power just 'cause some watery tart threw a sword at you! I mean, if I went 'round saying I was an emperor just because some moistened bint had lobbed a scimitar at me, they'd put me away!