Its a real problem. Insurance companies have a ton of weight by virtue of representing many patients, which they effectively leverage for lower prices. Those paying out of pocket have no such leverage, and pay much more.
Hospitals have leeway in what they charge. If you make it clear that you will be a pain in the ass unless they give you a good deal, they will do so. This is where you drop words like bankruptcy, $25/month for 30 years payment plans, and threaten to contest every single item (out of 47) on the bill. They will take all sorts of nickel&dime charges off your bill, lower prices here and there, etc, giving you a reasonable deal.
That was still a lot more money than I had at the time. If I wasn't lucky with good credit, I would still be in debt.