A market allows for innovators to find ways to deliver better service for less money and thus offer a more competitive product and gain market share. This competition forces all players to improve, or be forced from the market.
A publicly owned and operated business has no incentive to improve its operational efficiency and little to no incentive to respond to complaints in service.
In the US we have two examples of public insurance providers Medicare and the Veterans Administration (VA).
What we haven't tried as far as I can tell is to apply regulatory pressure with marketing forces. So for example if we made it illegal for pharmaceutical companies from preventing generics to be sold or marketed. If we required by regulation that all facilities publish their costs, and to whom those costs were paid, for every procedure. If we required that all medical procedure costs and options at every facility had to be made available at a publicly searchable database. Etc. It might help in making the market more efficient.
As to pharmaceutical costs, there's two things that need to happen First is that extension patent approvals must be significantly reduced (requiring moving more of the patent fees up to the application stage to cover more review/appeal costs). The second would be statutory licensing for all patents on medication after say 5 years, allowing for generics at a fixed patent licensing fee after the patent is 5 years old.
Those would be things to improve competition.