Why not just have a system where, instead of taxing people and spending that on a healthcare system, you let the people keep the money but have it stored in a forced savings account which can only be spent on healthcare, coupled with an optional emergency state-provided insurance plan that only covers the most expensive and life-threatening cases. The Government stepping in with subsidy programmes for the poorest, and specialised programs for emergencies (natural disasters, terrorist attacks, viral pandemics, etc.)
Let the savings be transferable between family, and inheritable, coupled with highest interest rates on the savings account, and you'll also have reasonable incentives to encourage your people to live a much more healthy lifestyle.
Rather than having some full-on national system, or full-on private, a nice little mix of the two could be the best solution.
It's the system that they use in Singapore. Only 3% of its GDP is spent on healthcare (2/3s of that using the savings plan and other private fundings, the other third comes from Government subsidies), and yet they have one of the best healthcare systems in the world - with the lowest infant mortality rate in the world, and those born will enjoy a nice, 80 year long life, the 15th highest in the world.