"Shopping around" for healthcare in the US is impossible.
Staff are not allowed to tell you negotiated prices. They don't even know. On top of that, medical billing will jack up the prices by adding extra billing codes.
Healthcare in the US is a scam. Without transparent prices it will stay this way.
That's certainly not true. Try it. I've negotiated with a lot of outfits who claimed their prices were not negotiable. That's just a crock.
A good friend of mine is a successful businessman, and certainly does not need to negotiate. But he negotiates everything - it's entertainment for him. I'm constantly surprised at the amazing deals he gets on supposedly "non-negotiable" items, including medical bills.
Your suggestion is laughable. We asked three different hospitals what the cost for our daughter's birth was going to be; none of them could provide us with the answer until after the claim was submitted to our insurance. The out of pocket from the hospital we picked was $12k.
Now negotiating a bill down after the fact? Sure, if you can't afford to pay and document it, it can be done. But if you can afford to pay? Be prepared to be raked over the coals.
US healthcare sucks, and your anecdotal evidence is weak at best.
And I can have the exact same test done twice and get different bills.