Indian Health Service, or even Medicaid in many states, supports the idea that a government health care program is necessarily better quality than through our current system. Would you want a health care program administered and funded by GOPers?
Heck even 30% of Medicare beneficiaries choose a Medicare Advantage plan run by private insurers under government guidelines. Medicare Supplemental policies and drug coverage are, you guessed it, private companies.
Would we be better without them, yes if we can ensure that GOPers never gain office again to cut funding, slant the rules to hurt the poor and sick, etc. Until then, I'm not sure we'd be better off with only government insurance. I do think we'd be better off letting people opt into a government program. I believe eventually, most of us would select the government program.
Also, while I agree most corporations seem to be profit maximizers, the law actually does not require that. It's a misconception --
http://www.nytimes.com/roomfordebate/2015/04/16/what-are-corporations-obligations-to-shareholders/corporations-dont-have-to-maximize-profits