There's no way that's ever going to work, though.

I disagree vehemently.

1) The choice of what to buy (and who to buy from) is primarily made by a different person than the one paying the bill.
2) The provider of the medical service has immeasurably different and better information than the person buying it.

No and no.
Many, many medical procedures are elective and non-emergent. Patients have choices as to the provider and schedule. Many patients also educate themselves on what each procedure is and how it affects them.

Blew out my knee in 1999. As it happens, the guy that does knee surgeries for the Colts/Pacers/Purdue/Indiana University did a once a week trip up to West Lafayette to meet patients. Dude has his coterie of fellows and med students with him. Since I had wrecked my knee and nothing better to do than pull MedLine articles up and ask him questions about the papers he's written. I got an impromptu anatomy lesson and he went through the whole thing step by step.

Does every patient do that? I'll admit, I'm more detail oriented than a lot of folks but that just means there's a spectrum of folks who WILL take the time to understand every aspect of the care they're getting.

If I (as the patient) am not paying the cost of the service, I am utterly indifferent to differences in price between provider.

Because you have no incentive to. However, if your insurance company told you there'd be a discount on your next premium if you picked Hospital A or Provider B you would take notice.

All this is infinitely better than an unwieldly government driven solution that seizes partial control over a quarter of the nation's GDP (or more).