All things being equal, the private sector will always run things more efficiently than the government because the private sector does not have a monopoly. For instance, consider going to the DMV. It completely sucks, the employees are usually rude or brusque, there is a byzantine system of bureaucracy, and you don't have any fucking choice, so just smile and take it.

Now, imagine if I set up a competitor DMV where I paid employees a little more to give you service with a smile that was efficient. If the original DMV wanted to stay in business it would have to shape up its woeful act. But, I'm not allowed to setup a competitor DMV. So, you get what we've got, which is a place populated by inexplicably surly employees (considering they get about 75 paid holidays a year) who move at the speed of molasses. What motivation does anyone have to make that place better, when the government forces you to use it?

There are exceptions to this rule. For instance, when the government subsidizes things that society depends on but are not necessarily profitable to individual outfits. Or, there are times when competition isn't possible (utilities in an area for instance). But I think to make your point, you'd need to specifically outline why the government would do better at running healthcare than private individuals (and I don't see anything like it - I'm imagining some "doctor's union" driving prices up further while simultaneously lowering the quality of care).

So, it's not that there's anything necessarily "good" about the private sector running it. The consideration here is how things get worse when the party running it has a legally compelling monopoly.

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"Government big enough to supply everything you need is big enough to take everything you have... The course of history shows that as a government grows, liberty decreases."

-Thomas Jefferson