I think you just described a government takeover of healthcare insurance. The money we pay in is called taxes, and some pay more than others. Where your plan diverges from government coverage is that the government will not keep a tally of what you have paid in and give it back on that basis. What your plan and government takeover does is cut out the middle man -- insurance companies. A lot of our healthcare money is going to insurance companies and admin workers (in the doctor's office and the hospitals) who have to deal with who pays, how much they pay, and collecting those payments, etc. I actually have no problem with government-run healthcare, IN THEORY. However, when the government starts collecting that kind of money (like they do in the form of Social Security and Medicare), there is a tendency for politicians to "borrow" it and squander it. Plus, any agency set up to oversee all that money we pay in, whether government or private, would have administrative costs and we would most likely be back where we are now. Insurance is not the answer to our problem, insurance IS the problem. We simply need to be able to set up our own accounts, and accounts for each family member, at our own bank and then somehow show it on our tax return, and maybe pay a reduced tax on it, or no tax on it like an IRA. For those who have no account, they would have to petition the government for help, if the medical attention was actually necessary. The funds that would have gone to all those bureaucrats would be used to help legal citizens below the poverty line. In other words, a system needs to be set up to only help those who really need help (similar to Medicare) and not a system that will upset healthcare for everyone. If doctors and hospitals have to deal with millions of tight wads looking for the biggest bang for their buck (the way they used to before the advent of healthcare insurance) instead of a disinterested bureaucrat or insurance agent, prices would go down.
However, the whole deal revolves around expecting people to do the right thing, and, sadly, many will NOT do the right thing, which is the problem we have today. If everyone would just get themselves covered with at least a disaster policy (high deductible, low monthly payment), much of our current problem would go away. Kicking out illegal immigrants would solve another big part of the problem. With your plan or any kind of "you take care of your own account" plan, many people would not sock money away unless they were made to do it. Of course, there could be a tax penalty if you do not sock the appropriate percent away, but then we are back to the government telling us what to do.
My point is, nothing will work that well unless we are all willing to deny help to those who refuse to plan ahead and help themselves. We must help those who cannot help themselves (as Medicare now does with the elderly), but we have to tell those who CAN help themselves and refuse to do so ... to go fuck themselves. They are the ones causing all this hoopla.
The whole problem I see is that insurance has caused the problem and we want to use it as the solution. I think stary-eyed politicians see all the money going to doctors, hospitals, and insurance companies and they cannot wait to get their hands on all that cash. They have no problem dipping into Social Security funds do they? Any time large sums of money are set aside for something, you can bet it is going to be spent. People have to be rewarded for staying healthy, not for getting sick. That is another major problem with all this: the people least able to pay are the ones that usually need it the most. The problems will never be solved if we think the solution is raising more money and making more money available to those in need. The healthcare providers will just raise their prices, which is exactly what has been happening over the last several decades. Supply and demand at work. People have been supplied with more money (via insurance) so they demand more frequent and better healthcare, which impacts the supply (there are only so many doctors), which impacts the costs. It is a viscious cycle. Imagine you are on an island and you can find just the right amount of food to eat every day to keep you in good health. Now imagine a large crate of food and liquor washing to shore every day or so. Are you going to sit on your butt, and eat and drink what is in the crate or go foraging for the local food? The natural balance of things has been disrupted by insurance money washing ashore on a regular basis. The only difference is, a higher and higher bill is washing ashore with it, but we still act like it is "free".