Earlier quoted context omitted.
I would not describe government as heavily involved in health care. The prices you see rising are from lack of meaningful competition and transparency allowing providers and insurance companies to dictate prices virtually arbitrarily and certainly irrationally to the consumer using the services. Meanwhile, government is the only meaningful access many have to healthcare and education.
About half of health care spending in this country is done by the government. That's heavily involved by any metric. Secondly, the amount of regulation of it is very, very heavy. The price rises are due to this. See https://www.theatlantic.com/magazine/archive/2009/09/how-ame... Note how prices for Lasik eye surgery, which has little government involvement, have trended down dramatically.
Either way, we need price transparency before people start acting rationally, and government regulation of irrational spending is going to be wasteful. Price transparency would make the promise of single payer (government) healthcare actually sustainable for a populace; without it, you’re just making the healthcare industry richer without actually providing a proportionate amount of care to the public.