You raise a lot of good questions and it can be hard to grasp how the police could operate outside of a government structure.

If you really want answers then I would say that the best thing for you to do is read the chapter of For a New Liberty that I linked to the other user.

In brief I think the biggest point that you have wrong is when you say we need someone to have the most power. All security organisations should have equal power. There is no need for a police with extra powers. You also make points on how police are held accountable but so often police are held to a different lower standard than regular citizens. Rothbard makes the point that free market security would be much farer and held to a higher standard than current public policing.

There would still be courts to hold security companies accountable they just wouldn't be monopolised state courts that can choose what to prosecute but would instead be private arbitration organizations.