[smoe]

How does this hypothetical world not end up with people that can afford to hire the more powerful protection agency be favored in those settlements?

Not that I'm happy with the current system, but I can't really see how a privatized one would not end up with an even bigger class divide where the laws are only enforced against people that are on the same level as you or below.

[jokethrowaway]

Because private court systems would have a reputation as well. You can also have 3rd party companies auditing and reviewing the system. If a court system is notably unfair, it will lose business.

I don't see why corrupting a private judge would be easier than corrupting a public one.

The private one at least could default if they had no more business due to their unfairness.