Or we could make it legal to open hospitals, or charge differential pricing based on the time of day... you know, basic stuff?
Most people have no idea how captured healthcare is. There's no competition, indeed most healthcare is government granted monopolies already.
Having lived in countries with "free" healthcare, that isn't all it's cracked up to be either. A good example of cost cutting: just make the disease illegal, like ADHD in the UK. If it doesn't exist, nobody can claim medication for it. Problem solved.