Prices are capped and regulated,
For all income brackets?
if a procedure costs more than it is worth to the insurance company, they will simply cut the provider out of their network and the providers know that.
Healthcare isn't something you can just order online (yes, you can order pharmaceuticals, but I'm talking about things like surgeries). It has geographical significance. I'm not certain that insurance companies can basically say "If you don't lower your costs, then you won't have any business with us." Their customers would either have to move away to a location that has caregivers that are accepted or they'd change insurance companies.
How is geographical significance dealt with, then? We do have a very large number of hospitals, but hospitals are also located to serve a population. When the population grows beyond their capacity, the hospital is improved, a new hospital is built, clinics are established, etc. While it's possible in cities with large numbers of hospitals that a population can just swap to another nearby hospital, that would completely screw with population metrics and you'd likely see horrible overflows. The hospital that refuses to negotiate down their prices enough would likely shut down, prompting a pretty horrible situation. Now, the hospital wouldn't want that happen so they would be willing to negotiate to a certain degree, but the power is not entirely with the insurance companies because the hospital can say "Well, if you actually went through with your threat, everyone would be pretty screwed and it'd come down to who could better convince people was the bad guy of the situation." This is to say nothing of rural populations which would be in considerable trouble should insurance companies refuse to cover local hospitals.
Furthermore the insurers can't drop people when their care becomes expensive, again something that providers know. So if the insurers don't negotiate prices down they don't make a profit. You can't ask someone to give an inch when they're back is against the wall.
This is true, but then prices will only be adjusted such that the insurers turn a profit, not necessarily that healthcare costs in the US lower to near the levels of other countries. Now, if prices have been capped and regulated at all income brackets, then that's a very significant facet of this entire situation and how healthcare costs can be significantly lowered comes into focus pretty easily.
You may not think these negotiations work, but Medicare has been doing them successfully for decades and Medicaid has been doing them even more successfully than that.
And yet healthcare costs are still absurdly high, showing that the negotiations from Medicare and Medicaid aren't nearly enough. They only deal with a fraction of the population, a fraction that hospitals are likely to make deals over considering it'd be good for PR and any losses incurred can be covered by the cost of other services to those not covered by Medicare and Medicaid. In addition to that, Medicare and Medicaid are publically funded, so grouping them together as being the same as private insurance companies is a bit dubious. Are you saying that all insurance companies are going to band together and blanketly demand that costs are cut across the board?