USA: Gov't run health care out preforms private care.

No, that isn't the case for Medicare at all (perhaps you should read a few studies that analyze the system), the reason Medicare isn't financially sustainable isn't because of fraud or the government's inability to run the program (it is half as expensive as private insurance) but rather it is because of high medical costs coupled with the aging population.  The government can provide excellent health care but the public/voters need to be willing to allow both the system to be established and a proper sustainable funding stream.  

And this would be where bringing in Government run health care for all would help, as the funding comes in from younger people's taxation for health and subsidises the treatment to the old....think about it in the way private insurance works but you are covering everyone therefore everyones money goes solely into it.