it would not because the margins in the insurance industry aren't high. The OP is entirely correct, insurers are a source of payment, not a driver of cost. Insurers don't benefit from hospitals charging them high prices, they try to negotiate cost down, that is their service. They don't enjoy paying hospitals more money.

What is driving the actual cost is the high expenses hospitals have, and the biggest source of that is labour cost, as in any other industry. American healthcare workers, doctors and nurses, earn about 2-3x as much as their peers in the developed world.

One of the biggest benefits of a national healthcare system for consumers is that the government becomes a monopsony buyer of healthcare workers. The British system is half as expensive because the government gets to set the prices for its healthcare workforce and that gets passed on to the patients.