Oh, but there IS a legal obligation... if you go the ER they must care for you. It is illegal for them to turn you away.
Which is not the same as a duty to furnish "unlimited care", as you put it.
Of course, I would expect most doctors to want to help... but if there are 50 patients in the ER and only 5 can pay / have insurance, I would expect most doctors to help them first (and possibly care for only 20 other patients for free that day because the feel like the others are not seriously sick and that their time is more limited).
We are not talking about selling chocolate bars here. If those 50 patients are not treated in the order of severeness, some might die. I would expect anyone, doctors and otherwise, to help those first who would die if nobody helps them. Everyone has that obligation.
Your solution would be excellent, if we decided that life and death don't matter. But if that is so we can also disband the military and wait for someone to come and kill us all. If we have a tax-funded military that kills people who want to kill any of us, we can also have a tax-funded clinic system to provide a similar service of keeping everyone of us alive.
Anyone who needs special care beyond basic survival within the safe environment of our society can hire their own militia or their own doctor.
This would make being a doctor more profitable, which will result in more people becoming doctors, less doctors retiring, the hiring of more support staff (nurses and the like) and the creation of additional support technology and equipment. Overall, more people would be treated and cared for, and more people will buy insurance (because there is an actual incentive to do so).
Idiots will never buy insurance because they are idiots, not because there is no incentive.
People will always think that that European holiday is more important than health insurance. Legislation that subsidises health insurance won't change that and neither will relying on the market to lower costs.
The only thing that ultimately makes some people do the right thing is force, which is why we shoot at terrorists and do not offer them cheaper targets.
If we force people to pay taxes to support the (necessary) military, we can also force people to buy health insurance and pay taxes to support a basic healthcare system of free clinics (which we need for training purposes anyway).
Without basic healthcare available for free, diseases will spread. That's something the free market simply cannot deal with.