The police are state-controlled rather than community-run, following state laws not community rules.

And as the results show, empowering the community to defend itself works out much better.

How is a local militia of citizens different than the police?

The police is unaccountable and above the law.

The police structure is not strictly local, that's an assumption I wouldn't make. Here's a reddit thread (I'm assuming mostly USA responses) with many verified officers saying they live half an hour, an hour or more away from their jurisdiction: https://reddit.com/r/ProtectAndServe/comments/1c59je6/how_far_away_do_you_live_from_your_department/ - "Seems most LEOs I've met don't live in the same area they work"

Someone in another thread says to an OP: "I am assuming you are in the US, but elsewhere it is somewhat unusual for a police officer to live within the jurisdiction they police. Often you will find particular towns where officers might cluster, but living in the same part of a city, or in the same small town, is undesirable due to familiarity and reprisals. Obviously there are exceptions."

A local militia of citizens will tend to be more familiar and accountable to the community they work within than some strangers working for the state, who are alienated to the subjects of their law enforcement. Plenty of systemic issues with police are a result of this alienation from the society they police.

In this example of Cherán, IIRC the corrupt police they evicted were municipal police, not local. By being members of the community, the local militia that replaced them could resolve more issues socially rather than by violence (inc. threat of violence) and were not accountable to the national government, but instead to a local government directly elected by citizens. So what this means is, unlike most police forces, these law enforcement agents are accountable to the community they govern. This reduces the tendency for corruption and abuse, and makes the job less enticing to those would seek to abuse it.

Police are frequently not citizens of the place where they work. I work for a small city in the Development department, and not a single person at City Hall from police to court to administration has ties to the city itself.

In our case it's because the tiny city is an enclave for the mega-rich and you literally can't buy a house for under 3 million dollars right now, and the new-build homes are between 5 and 30 million, and they're not gonna pay staff enough money to live in town.

But even in other places, lots of police live outside the city where they work, if for no other reason than specifically because they don't want to enforce the laws on their neighbors.

Certainly not.
They are not loca and they do not serve the interests of the local people, or any people for that matter, nut only the interest of the state.
In reality the few people who control it.