A fun scenario, where you'd need local, armed police to have never dominated law enforcement in the US. Pretty much the only POD I can see is that Sherman gets his wish and the Posse Comitatus Act of 1878 is never passed, allowing the Army to become a peacetime Gendarmerie and enforce civil rights.

On top of the Klan staying dead, Jim Crow never being enforced, and the Constitution & Federal Law being enforced over all others (granted, this has its own drawbacks), cities would never have a reason to adopt local, armed police departments.