The issue is not that health insurance companies make too much money (ok, it's not the only issue)They, along with the system they put in place introduce immense amounts of friction into every medical interaction and prevent doctors from practicing good medicine.
You're giving doctors a little too much credit. While most of them have good intentions and try to act in the best interests of their patients, something like a third of the care they deliver is considered "low value" in that it's not evidence based and isn't likely to benefit patients. While some of the friction caused by health plans is just pointless waste, the utilization management processes can actually nudge doctors towards practicing better medicine.
https://www.bloomsbury.com/us/price-we-pay-9781635574128/
The insurance companies are not stopping the government from paying for everyone's healthcare. It's the other way around, governments are using insurance companies (better referred to as managed care organizations since they don't really sell insurance) to add the friction so that some people get more healthcare and some people get less.
Who gets more and who gets less depends on who has political power (that's why the old and non working get subsidized by the young and working), and in a democracy, this question ultimately comes back to the voters.
Bottom line is due to demographics and restrictions in the credentialing process (including for medicine itself, one of the costliest components of healthcare), there is nowhere near enough supply of healthcare relative to demand, AND due to the enormous damages awarded in lawsuits in the US, the cost of liability protection is sky-high and increases prices for every step of the healthcare chain.
We need way more healthcare providers, and tort reform, and publicly funded medicinal trials, and without that we will continue to limp on with this bureaucratic maze to essentially reduce demand to manageable levels.