I don't think profit-making businesses should be in charge of healthcare, and obviously my preference is that Goverment wouldn't either. In fact, i have my own ideas about a non-state system i call LHS but i won't bore you with the details. My main concern is that everyone actually gets treated.
Why? Because profit making is evil? What is profit making, exactly? Profit making means that the people that run the business make their living from the business, their livelihood depends on the success of the business, and they are also responsible for the business and its employees. If they don't provide what the customer wants, the business suffers and the owners suffer directly. If they don't keep the employees happy, the business suffers and the owners suffer directly.
Profit making does not mean illegal, deceitful, or corrupt.
Finally, the idea that Government-run systems do not allow free choice is fallacious. There is often a wide choice of doctors.
This is not true. In government run systems, patients are limited in their options. There are all kinds of gatekeepers, you have to get approval to see specialists or to receive certain tests or use certain diagnostic equipment.
Canada is a great example. Do you think all those people waiting for treatment are doing so voluntarily? No, they are prevented by the government run system.
And, there is availability. In some countries (Germany, Japan, Taiwan) they are facing a growing problem with a lack of people working in the medical field because of government imposed restrictions on pay. Who wants to go to school for years and years, work long hours, and then not be rewarded appropriately? You can be a doctor, or be an engineer and get paid what you are worth (more than a doctor). Not a hard choice for many people. So, if there are too few doctors, your choices are limited.
Its not just physicians either. Government run systems keep costs down by restricting reimbursement to hospitals. Half of allhospitals in Japan are in the red financially.
Government run systems claim they do not limit choice, its a nice phrase, but its false.