I've heard the exact opposite - you can negotiate a better price paying cash vs having them deal with insurance. Maybe that's only if you're dealing with smaller clinic or a specific doctor though? I have no personal experience doing so.
Disclaimer: Uncle Sam has covered me since I was a young adult under Tricare. My brother, though, spent some time without insurance.
It is indeed if you're working with a smaller clinic or a specific doctor that you can often
negotiate the lower rates with if you pay cash
up front. Many, however, won't even see if you if you don't have insurance or pay up front. My brother did manage to negotiate deals of up to 50% off of what insurance would have ultimately paid. Some will work with you, some won't.
Meanwhile, hospitals and larger clinics have complex deals with insurance companies, can't figure out your bill until well after you've been discharged, and won't accept cash up front even if they could figure out an estimate for the services you are going to receive. It's a mess.
So as long as you can stay out of the hospital, are willing to shop around and yes, haggle a bit, you can get healthcare cheaper than what insurance pays.
The problem is what happens when you end up in a hospital anyways, don't have negotiating power until you're threatening bankruptcy, and the hospital charging you double of their average insurance payout is the best you can hope for.
Anyway if you have to mandate what they charge in a specific situation, there's probably a regulation or requirement elsewhere in the system that needs addressing.
Treat the root cause, not the symptom.
I see the problem as that, unlike nearly every other industry, they're not required, by market forces if nothing else, to post prices or give estimates before providing services. Not even hourly labor rates.