I contend that healthcare is a human right. A fancy car? No. Any car? No. An xbox, a fishtank, your favourite china pattern on the table, or an all-in drinks-included tour of the Caribbean? No, no, no, and no. But the ability to call upon your community for help in getting well and have that happen not by hoping, not by asking, not by begging, but as a matter of course and not chance, that is a human right and a hallmark of civilisation.
I think you're confusing rights with obligations. It reminds of me my niece when she was little; she would just pick up some toy or other item that someone else was using, and when there was protest, say, "You have to share". The concept of sharing was used as a justification for stealing.
Rights are what pertain to a person by nature. The first one is to be in charge of my own life, which on the flip side means not to have anyone else be in charge of my life. That flip side means that anything which puts me in charge of someone else, or someone else in charge of me, is not a right, but a violation of rights. So healthcare cannot be a right, because it puts the needy in charge of others without asking their consent.
Healthcare is like sharing: it isn't a license to steal, but a demand on the individual to do good for others. I can't say to another person, "You have to take care of my health" any more than I can walk up to someone with a nice laptop and walk off with it on the basis of "You have to share". Sharing is a moral imperative to those who have, to do for those who do not, as is healthcare -- which is, after all, a form of sharing.
It isn't the business of government to mandate moral positives, only to penalize moral negatives, i.e. harm. Government not only can but must say, "You shall not steal", but it cannot and must not say, "You must share" -- because in the moment it is mandated at the point of a gun, it is no longer a moral positive; it is no longer sharing, but extortion.
Your ideas about lifting restrictions on medical school places are sensible with regard to cost, but still say nothing about universality. On the point of cost, there are so many examples of health care costs going down when they are shared broadly across the entire population, and moreover not diverted to insurance claims administration/processing, marketing, and profit margins. If you don't want to see it, fine. If you struggle to pay for your own health care then at least there is a certain kind of natural justice in that you are struggling at the hand of a system you're content to support lest anything else be an affront to your liberty. Even though countries regularly accomplish this with freedom, liberty, free speech, public scrutiny, informed, literate populations that can make choices and give their consent and broad support for public universal health care.
My health care comes without marketing or profit margins, so I'm not sweating that. Why more people don't join, and why more don't set up, similar systems has always puzzled me. Mine comes through a church, which handles administration rather cheaply... and to which I from time to time protest that it isn't offered to all members instead of just those who have been in some capacity "church workers" somewhere along the line. To me, a church is a natural organization to be offering health care under the power of "broadly sharing" costs, but it doesn't seem to happen. Maybe there's some government regulation against it?
Show me a country that doesn't require everyone to pay for that universal health care under threat of men with guns, and I'll concede that they have a system which operates with "freedom" -- until then, you're blowing smoke. As MLK pointed out, unless all are free, none are free, so if even one is paying only due to the coercion of force, there is no freedom.
No tanks, no gun-toting soldiers in the streets, no hyperbole, no drama. Just showing up at a doctor and getting diagnosis and treatment. However, you haven't answered what the 17 year old is supposed to do on his 18th birthday when he doesn't find work that will get him into a physician's care. I'm prepared to do my share for that person, and I infer from your posts in this thread (unless you suspected him of having some unworthy disease like maybe a smoker with emphysema) that you would too. But if you lacked that character, withheld your taxes or worked under the table so you could do your best to ensure the 18 year old kid didn't get to see a doctor, I would be glad to see you coerced by as many government jailers as we could muster.
If people don't want to pay for your health care are required to do so by law, that is by definition at the point of a gun -- you don't pay, the goons come for you.
The 17-y.o. should plan ahead. And in a civilized society, outfits such as churches, lodges, brotherhoods, or whatever would already have made provision for this. I wouldn't mind mandating that everyone be required to join some organization which provides care for all its members, whether it be the Elks, Masons, Boy Scouts, or whatever... and in a civilized society, I don't understand why such organizations don't offer such coverage. In my vision of a properly civilized society, everyone would join a sort of lodge/fraternity/fellowship at age 16, until then being covered by that of his or her parents, and it would be the responsibility of the lodge/fraternity to see that the person is cared for when needed, as well as kicked in the ass when needed. These organizations could have their own doctors, etc., contract with outside organizations, and/or provide payment to whatever provider a member-patient chose.
And if any individual decided not to join such an organization, it would be with the knowledge that he or she would be utterly without claim on anyone else for health care, or any other benefits (there could be many) provided by them.