The problem is not lack of health insurance...it is the cost of health care. The cost of health care is driven up by new technology, but also by increased gov't mandates and by private insurance companies skimming money off the top (when consumers pay their premiums). Also, when you have people who don't have health insurance of any kind (either because they can't afford it, or because they chose not to buy any...and there are people in the latter category) and they get some sort of severe illness, cancer or get into a major traumatic accident, they get care anyway, although they can't pay, and all the rest of us foot the bill for that via higher health insurance premiums. We also are all paying for the high, high rates of obesity, smoking, etc. in our society because those unhealthy people drive up the cost of health insurance.
I'm 34 and when I was a very little kid, and my folks took me to the doctor, they just paid for it. We were not rich (at all). We did not have health insurance until perhaps the early 1980's or very late 1970's...per my mom, most people didn't have health insurance, or at least a lot did not. This is because you pretty much just paid for your medical care...at first I was kind of incredulous but then she pointed out that before the 1980's or so, if you got some bad kind of cancer or you got in a really bad car accident or something, usually you just died because there wasn't the really advanced medical care that we have today. It just didn't exist. Also, if you look at things like heart attacks, back in the 1970's or so, they had beta blockers, some aspirin and probably nitroglycerin, but none of these other fancy medications we have now. All these new things cost money. There isn't an easy solution to this conflict between cost and quality...if there was an easy solution, this would be fixed already.
Personally, I think one thing that would help was if health insurance didn't pay for every minor little thing (except in the case of indigent patients). Health insurance should really be for catastrophic events, such as major trauma, cancer treatment, big surgeries, etc. That's how insurance for your home, auto, etc. is. If patients paid more of the true costs of their care, they would look for a better deal and people would not go to the doctor so much for a runny nose, etc. I see this a lot because I do primary care. I don't mind taking care of them, but I think some folks come in for minor stuff because they are only paying $5-10 and just want a note to get off work.
I do think there are a lot of docs and nurses, etc. who volunteer their time for free to take care of the uninsured. I do this a couple evenings/month, and we can take care of a lot of minor stuff that otherwise people might decide to go to an ER to get it taken care of.