There's nothing stopping you from starting your own non-profit health insurance company. If greedy health insurance companies are really the root of the problem, you should be able to out-compete them fairly easily.
You are correct that health insurance companies aren't the problem here. But there are a lot of things stopping you from starting your own non-profit health insurance company. Medical insurance is heavily regulated at the state and federal levels. Launching a new one takes years of work just to get through the paperwork and legal compliance issues.
