As soon as there are people who opt for first-aid type health care combined with boosting their immune systems and a healthy lifestyle, the market will respond. I'd wager that after ten years or so of that option, people who choose it would be just as healthy as those who choose to be slaves to the health care -insurance - big gov "solutions".
Your system’s fine, except you need to make a decision about those who choose not to buy insurance, then suddenly need health care but can’t afford it. Care is provided by people, and those people are entitled to be paid for their services, so you can either stick the taxpayers with the cost for it, indenture the family, or leave the guy laying in the gutter.
The last option’s a hard sell, so the most likely solution for that situation is some combination of charity and teaching hospitals, which was the foundation for Medicaid back in the day.
Maybe you could build in a low cost catastrophic policy that those who chose to go the first-aid route could buy. They’d have to get really sick to use it, but it’d keep them out of the county hospital if something really expensive happened.