I think the big problem in the US is not just the insurance companies themselves but that health care is so luxurious it is ridiculously expensive.
If I went to a top London private hospital to give birth paying everything myself it would still be a fraction of the cost charged in the States.
For example, here are the costs of giving birth in the Lindo Wing at St Mary's: https://www.imperial.nhs.uk/privatehealthcare/hospitals/lindo-wing/lindo-wing-prices/index.htm
Most other private hospitals charge less than that.
Or, a more simple example. I was unfortunate enough to need to go to A&E and then have a D&C in the US. My travel insurance had to pay more than $20,000 for the pleasure. Yes, the room was nice and I had a remote control for the blinds but did I really need such luxury? The en suite was the size of a small apartment!
In comparison, I have private insurance in the UK (costs about $170/month), which covered the same procedure in a private hospital here (still luxurious but without the remote control and my en suite was smaller). The hospital charged my insurance the equivalent of about $3,500. I had my own room and the food was better, too.
Had I had a D&C on the NHS it would have cost even less (nothing for me to pay, of course). The only downside would have been sharing a room, which I have done often enough and it really isn't the end of the world.
No wonder insurance in the US costs so much. It has to pay the hospital's extortionate bills!