For literally decades the US did not open a single medical school due to lobbying from the medical industry. Even though we've tried to reverse the trend since the 90s, there are still fewer medical schools today than there were 100 years ago. When the government decides how many doctors there will be, it may get the number right or it may get the number wrong. But that's nothing akin to "free market-like." And it's a major part of the explanation why US doctors make 2x+ what their counterparts in countries like Germany make.
During the New Deal era regulations wage caps were put on workers in the hopes of staving off inflation. Employers, wanting to entice good employees to work for them sought to get around these wage caps by offering health insurance. This then spread like wildfire, coincides with the time healthcare inflation separated from general inflation, and began the trend towards a lack of price transparency.
Most of the differences in outcomes between us and other wealthy nations amount to lifestyle/culture much more-so than affordability and availability. Americans are very obese. Not having to pay a copay to visit the doctor is not going to change that. But obesity means we'll have more problems with newborns, lower life expectancy, etc.