The US healthcare system is deeply flawed but only a small fraction of spending goes to emergency care involving ambulance transportation. The vast majority of healthcare spending is for elective services and patients do have time to shop around. Self-insured employers have been cutting costs by pushing employees to high-deductible health plans with HSAs, which gives plan members a financial incentive to find cheaper options. Of course there are still challenges around getting meaningful price estimates from providers despite recent federal regulations on this topic.
In fact it's almost impossible to get a price estimate up front in many cases. A procedure may involve multiple codes which may or may not be covered by your insurance, and the only way to find out if they are covered is to have the procedure and then check the bill. Frequently you won't even get the codes up front. If you call they can't even say if a particular procedure will be covered or not, the call centers are just not set up to do that. There is incredible complexity as the procedure may or may not be covered depending on patient, provider, plan, how much they've spent this year, region, if the doctor's front office made a clerical error at any point, if the insurance company made a clerical error, if the claim is processed on a Wednesday vs. a Friday, etc... If you have a procedure done twice the coverage may be totally different on the second time, with some parts covered that were previously not covered and vice versa.
Thankfully you can often get denials reversed after the fact by calling and complaining, but that takes an hour and is another roll of the dice. The only people who like the US system are wall street people who own stock in the companies.