First name an example of a private monopoly so we can debate one that truly exists and isolate the discussion to that one thing
Is there any reason you want to isolate it to one thing? Does that benefit the discussion in any way? Were you trying to project that you don't believe anything such as a modern monopoly can be said to exist?
Monopoly comes in many forms, and only the most limited version of the argument puts it at "100% market control," whereas, if you read the many good laws that were passed in the wake of consumer and market harms, you'll see there are four specific elements, any of which, are not only a criminal act but are widely considered "monopolistic behavior."