I’ve always suspected that that incentivized health insurance companies to let hospitals inflate costs. If the government forbids me from increasing my percentage of the pie and I need to increase revenue I have to increase the size of the pie.

Yes, this and vertical integration. The insurance company may have limited profit, but if the hospital doesn't, and they're both owned by the same parent company, then the prices "inside the control volume" can be whatever fiction is most convenient to report to the government.

How many insurance companies are under the same parent org as hospitals?

Looked up the largest health insurer, they operate hospitals:

> Kaiser Permanente operates 39 hospitals and more than 700 medical offices, with over 300,000 personnel, including more than 87,000 physicians and nurses.

https://en.wikipedia.org/wiki/Kaiser_Permanente

The second largest also owns Optum with 60k doctors:

> Optum Care is a family of 60,000 doctors in 2,000 locations nationwide. We work together to help 20+ million people live healthier lives.

https://www.optum.com/en/about-us/optum-care.html

So I'd ask you back, how many insurance companies aren't under the same parent org as hospitals? The 2 largest are, wouldn't surprise me if most are.

I’m not sure. I don’t know much about healthcare in the United States.