Medicare’s overhead and administration costs (as a percentage) are actually a small fraction of the overhead and administration costs (and huge profits) of private insurers. Depending on how you calculate it, only 3% to 7% percent of Medicare money is spent on overhead and administration. The rest goes to paying for medical care. Insurance companies have overhead and administrative costs running in the 15 to 30 percent range.
Most things the government pays for are things the private sector isn’t all that well equipped to do. A private road system would be a messy, wasteful, redundant jumble, for example. Sort of like our current healthcare system. Conversely, there are things government would really suck at and it should stay away from. Making and selling TVs, for example. And while not getting a new plasma TV won’t kill you, not getting healthcare just might.