If the monopoly is not run for profit, it can. Executives of health care insurance companies take home bigger bonuses than bankers, and spend more money in lobbying than the oil industry. All of that was money that was supposed to be going to health care.
The U.S. spends 20% of it's GDP on health care, while most other first world nations spend about 12% and cover more people. Many European countries combine public single payer with public and private health care providers, and this seems to deliver the best results for the least money. And the 20% in the U.S. does not include the cost of medical bankruptcies, or the cost of lost opportunities caused by people who are afraid to strike out on their own and start their own business because their health care is tied to their job, and entrepreneurship means going without medical insurance until your business is a success. The American health care system strongly discourages entrepreneurship.
Meanwhile, companies that want to hire the best are saddled with the additional cost of expensive health care insurance. Even with the dollar at par, Canadians are cheaper to hire because the company doesn't have to carry this cost. And because private health care operates with small sample sets of people for a single company, actuarial tables don't work, because outliers are more frequent, so the companies demand more for insurance than average rates would suggest. The best way to bring costs down in insurance is to insure the largest number of people possible and spread the costs more evenly. This is precisely what public, single payer systems do. And they also do not result in a balkanized health care system, where your insurance may not cover you at the closest hospital, or even in the town that you happen to be in.