> Provide an incentive for companies to regulate each other
Do you think there is a reason every modern democracy resists privatising criminal prosecution? (They had it in Rome. We probably need a history of law section in the basic high-school curriculum.)
The justice system in this hypothetical would not be private, it's just that companies would be trying to sue each other for breaking the rules, accidentally advancing the public interest.
Provide an incentive for companies to regulate each other! Let rational self-interest clean up the market while the state steps back and looks on!