This is easy. Rich people control the money, not everyone. Particularly people at the heads of businesses, that could benefit from having their competition impeded by the law.
That's not entirely accurate. The Fed controls the money. Congress has ultimate power over the Fed, and the democratic majority has power over Congress. They've voted politicians in power who have been redistributing a disgusting amount of wealth to the rich for a very long time. One way Congress does that is, as you said, using government to limit competition in the market. Corruption would not be worse in a free market because competition cannot be impeded by force - only the state can do that.
I do believe there's corruption in the current system, the one you're suggesting would just be much worse.
No it wouldn't. Do you think customers are going to continue funding corrupt justice systems that allow people to be above the law? Competition prevents that from happening. The state already picks and chooses who is on the receiving end of justice and who is above the law. I don't see any bankers responsible for the '08 financial collapse in jail. I don't see any oil tycoons responsible for the BP spill in jail. I don't see any Presidents in jail for war crimes. I see a lot of poor people in jail for victimless crimes (like possession of marijuana) though! If you think that the elite don't have the ultimate control over who is above the law and who isn't in your monopoly justice system, you are competely delusional.
If I had the freedom to choose the justice system to finance, you can bet that I'd be financing one that would go after all of those criminals. And I'm sure many others would as well.
Explain how competition keeps corruption in check
Because people voluntarily funding corruption out of their own pockets are far less likey to tolerate corruption than people who are funding corruption because they've been held up at gunpoint. Now explain how monopoly keeps corruption in check. Good luck.
How would two police departments competing to arrest people result in a better system of justice.
They'd likely work together to come up with the most cost effective solution. It would almost always be too expensive for private competing police forces to go to war with each other over arresting people.