During the 30's it was suggested by Democrats in Congress that the govt provide health care and general pensions for Americans.  Business, at first, kinda supported the idea.

However, that support dried up when the unions started winning those concessions via CBA's.  With no other place to get such things, workers and unions went right to the company.  Again, the idea that standard of living issues and work/jobs were tied together.  In Congress, sad but true, Republicans had sided with southern Democrats to block this plan.  Even FDR feared such legislation would end up costing everyone politically as southern Democrats would do anything to stop 'radical' unions (even then, there were problems).

Business began to back off when they saw an opening.  By dealing with such issues on a case by case basis (via each business, etc), they could possibly win a few in court (there was, even then, a strong GOP/business view that such benefits were illegal and would be tossed out when taken to court).  By running a govt sponsored health/benefit program, all business would HAVE to chip in, much like they do today for SS or Unemployment.  At least this way, they stood a chance.

THey lost, of course, and since then unions have filled that role.  And, it worked, as long as business HAD to do it, or face strikes.  Or couldn't bail out of, say, Michigan and move to, say, Alabama.  (One of the first major GOP post war reforms was not pro-labor, but the anti-union 'Right to Work' bill, which gave us today's breakdown between states where unions can have a closed shop, and others where workers don't have to join-Every southern state is a 'Right to Work' state.)

Today, and even then, really, only govt can run such a program.  Nothing else will work, period.  The main reason?  Because the govt can manage it, and, pass laws to keep it effective.  If not, the first thing the GOP/business will do is pass laws designed to 'privatize' it.  Just like Bush is trying to do today with Social Security.