There is no competition now. That is the problem. No one selects their hospital or medical provider based on price.
Point me to a competent and working system that doesn't have the element of some competition in it.
That is mostly true. For some elective procedures, patients can shop, but lots of the very expensive care is urgent and/or complex, and there is no real way to shop for it. Try even calling your PCP or local hospital lab to ask how much a certain blood test or imaging procedure would cost you out-of-pocket: many times they won't be able to tell you. How can you shop if you can't check prices. For competition to result in increased quality and cost, there has to be true choice for the consumer.
The other side of this is that health care is regulated such that it can't work like other businesses. If you show up sick or in labor at an ER, and have no money or ability to pay, they cannot turn you away by law. If you have a stroke and your insurance doesn't cover acute rehab, the hospital must keep and rehab you (at your loss and usually theirs) until they find someplace appropriate you can go. They can't just apologize and discharge you.
If you show up starving at McDonalds, on the other hand, they don't have to feed you regardless of your ability to pay.
And in the ER waiting room, most of us would be upset if the guy with the nosebleed can pay extra to be seen and treated before the guy with the broken leg.
No question the system is a mess, and more discussion and transparency will help, but I think that most people would be uncomfortable if health care were provided and regulated just like any other business, and that makes it harder to redesign the system. I haven't seen these kinds of ethical issues enter the public discussion much beyond the basic "right to health care" that people refer to somewhat abstractly.
I work in an academic health care institution, so I have a vested interest, as we have and will always "have" to treat people with or without insurance, Medicare, Medicaid, etc. I don't pretend to have many big answers, as I am far from an economist, but I do think that our stats as a nation in terms of longevity, infant mortality, overall health measures, etc. are sad and shameful given our potential.