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 not

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.