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.
Well, there's a couple of problems here:
1) People who fail to pay their bills cause financial losses, which have to be covered by the hospital. Therefore, given identical personnel/equipment/etc costs, a hospital in a "poorer" (= more unpaid bills) area will always be higher in cost than a hospital in a "richer" area, since losses have to be distributed among the fully paying customers.
2) You cannot just compare different hospitals on pricing alone. Factors that make this impossible include rent, personnel costs, and equipment costs (e.g. a hospital which uses an old MRT will operate at a cheaper rate than one which buys a new MRT every year).
3) Medical procedure costs can not be estimated at all, given just the diagnosis. Right now my cat is at the vet hospital, turns out the (relatively simple) laying of a pipe to the stomach (she doesn't eat/drink anymore due to liver inflammation, and syringe feeding a cat is cruel for both owner and cat) isn't so simple at all for this unique cat. (For the fellow cat owners, the only thing not healthy about this cat is the liver. The rest of her is in a pretty good condition, and the vets believe she'll make it and purr on for years).
Regarding #3, I find that peculiar. Yes, for major operations there are always potential outliers, and you need to give a range and there may be potential large exceptions. But e.g. UK private providers will happily give you a quote in advance for most non-emergency care, and for anything with a straighforward diagnosis they will be willing to give you a fixed fee quote with certain exceptions. Not least because they have the volume to either self-insure, or get insurance to cover exceptions.