The US has these entities too - they are called "insurance companies".
As others have commented, people don't comparison shop even if they could.
This is simply false. People with low deductible insurance don't comparison shop because they have no incentive to do so. I have no insurance (my startup is minimally funded), and I do comparison shop.
It's a bit like having no idea how a car works or what could be wrong with it, and then being asked to pick the best mechanic and strategy to fix it.
Here is what I did a few weeks back. I went to a doctor who ordered an MRI. I called around, discovered the price of MRI's varied from 5-7k (INR, not USD), and then picked the closest one to my home (the price difference was small enough that I didn't care).
When it came time to get treatment, I did research on the internet, as well as asking three separate doctors. They all independently had the same recommendation - microlumbar discectomy on L5-S1. This was fairly apparent based also on the pattern of pain I was suffering, and just by looking at the MRI. I ruled out one hospital because it seemed dirty, chose a doctor/hospital to have it done (not the cheapest, but not the priciest), downgraded from a deluxe room to a private one [1], and got myself fixed up.
This doesn't work for emergency medicine, but most medicine is not emergencies.
[1] The default over here is to give white guys the gold-plated options.