Here in the UK we have the NHS which is great. But we ALSO have private insurance / hospitals / healthcare that is reasonably priced.
Costs for treatments are standardized and agreed on nationally: you can go and look them up. Your bill has codes on it that match this list. It is called TARMED: https://www.bag.admin.ch/bag/fr/home/versicherungen/krankenv...
Insurers under the LAMal system are restricted in what they charge for premiums too, and must negotiate this with the confederation.
There's still a lot of scope for private companies to make money here. LAMal covers what are deemed essential treatments, which is most things you need a doctor for. If you want 'alternative' medicine there are insurers you can pay more money to for that. Likewise, if when you go to hospital you want a guaranteed private room every time, then you can pay extra for that. And so on.
Even on the basic system, you have a choice between telephone doctor for all appointments except emergencies, family doctor for the same or the unrestricted system where you can ring up a specialist yourself without referral. Needless to say, the one where you ring the insurer first is cheaper, followed by the family doctor (similar to the UK: referrals are controlled via your GP).
I don't see any poor doctors here, or poor insurers either (insurance, behind banking, is a massive business here). So it is entirely possible to design a system that includes a large amount of private insurance that also doesn't randomly try to bankrupt people for getting stitches.
Also, in a country with both, the private system can't be any worse than the public system because it has to compete with it, so the fact that the UK has good private healthcare doesn't necessarily mean that it will be easy to fix the private system in the US without the existence of a nationalized system as well.