You just need more competition and regulations to make prices public and stated before the procedure. I live in a country with single payer healthcare. It sucks tremendously. Instead of paying there is waiting, sometimes for long months for absolutely crucial procedures. Fortunately I also live in a country with reasonable regulations and sizeable private healthcare facilities. I pay reasonable prices (which means often 5x-20x less than in US) if I need treatment and I can get it in timely manner.

It's not single payer that makes our system better. It's competition and reasonable regulation that makes it possible to have cheap services available.