Having lived in two countries that have free public health care at a good level, I can say that starting my own business is much more relaxing in that environment. I would not dream of doing the same in the USA at my age or family situation.
If the U.S. government stops incentivizing the connection between employment and insurance, starting a company wouldn't need to have any effect on your health care. No single-payer system required.
Also, you assume a slow in advancement in medicine. This can't be further from the truth. Small example: all of the patients registered in the UK (i.e. residents) are getting digital patient records.
Americans will be covered rain or shine once the healthcare reform bill kicks in. If you can afford it, you have to by insurance. If you can't, it will be subsidized so you can afford it. That's the goal, anyway.
Markets require choice to function properly. I think the American approach is less likely to result in unintended consequences.