You should talk to Doctors. They seem to have a quite different opinion of Medicare
You should also look at world-wide comparisons. Medicare and other public healthcare programs in the US account for more dollars per capita spent than all or almost all universal health-care systems in other countries, and deliver lousy results comparatively.
Canada--with our nominally single-tier, public, single-payer health care system--has longer life expectancy, lower infant mortality, and better outcomes by any number of other measures. Critics (sometimes justly) focus on input measures like wait-times, but at the end of the day what matters is that we are getting health care and getting good outcomes. We aren't even the best in the world--just middling-decent as these things go.
So the real question is not "why can't government launch a website" but "why can't the US Federal government, alone amongst all governments in all developed nations, provide a reasonable level of basic, universal health care at costs comparable to those in every other developed nation on Earth?"
This isn't a "government" problem. It is a uniquely American problem, and the solution does not lie in any general ideological fix, but in the detailed structure of the specifically American, particularly broken, Federal government.