It does if the government is not involved.
Capitalism creates bubbles & winners (aka monopolies). Government has to be involved to keep an even playing field and to smooth out the bubbles. No industry in the US has had no government involvement. All industries are involved with government in some form - if only to enforce contract law, or for more physical matters like using public roads, buying electricity from government run electrical companies, benefitting regulation that requires phone providers to provide service, etc..
What's being described here is what anarcho-capitalism and pure free markets always devolve into - the mafia. You don't get to just shop around for a better mafia if you don't like the one running your town.
Monopoly on violence requires enforcing that monopoly. Such a firm would also stomp out its competition. You’re describing government-run healthcare by an unelected government.
Government run healthcare, the healthcare where you just go to the hospital and then you just get treated.. (?) Government run healthcare does not have a great reputation for doing cutting-edge health care, but on the other hand it does have a good reputation for taking care of people for basic needs. How is that describing a monopoly on violence? Repairing broken bones is the same as breaking bones? (If the implied argument is that health care is so heavily rationed, that people are left untreated, tantamount to violence, I would beg you to look at the statistics of which countries have more complications due to untreated illnesses (hint, the US is high on that list [1]; or we could look at health outcomes per dollar spent, US is very far down on that list, US spends multiples of other countries for the same outcome))
I would think a more apt description and example of a monopoly on violence would be narco-states. Otherwise the example is a well functioning government, and that is demonstrated in the police force and judicial system (not healthcare).
When considering the absence of the government monopoly on violence, drug markets are the closest thing there is to a true free-market. And yes, there contract enforcement is contingent upon a bunch of people standing behind you with guns.
[1] https://www.cbsnews.com/news/medical-care-costs-americans-sk...
Or do you mean an unelected government has a good track record of meeting basic needs as well?
For the good track record, I'm referring to governments like the UK, Norway, France & Canada that all have better health outcomes compared to the privatized healthcare system of the US. The US system tends to stand out for the very specialized cutting edge (and super expensive) procedures; though you still have to overcome a lot of accessibility problems to get that. At present day, if you break a bone, have a child [1], need to be in hospital for a few days - generally you're better off not in the US
[1] "Despite spending less on coverage, France has comparable or even better health care outcomes than the United States. The United States has a higher rate of infant mortality. In France nearly four children die out of every 1000 live births. In the U.S. that number is closer to six." https://healthchoiceofmichigan.com/lectorium/MHzUCToycks
[2] https://www.brookings.edu/wp-content/uploads/2016/06/dutton.... This resource sites that France was ranked #1 back in 2001. While the US might have improved in that time, it goes to show it has been decades where the US is not the leader in healthcare. This implies that on average and for at least a couple decades running now, there are better outcomes in other countries (than the US). The dependent variable does seem to be how healthcare is financed, payed for, and how the profit incentive compares between those different places.