I disagree that we could have a privately run military. But, given that in many countries the state has provided the armament and equipment, not private industry, I think it's an important point.
The values of a good soldier are duty, obedience, order, discipline, courage, and self-sacrifice. These are not free market libertarian values. They are authoritarian values.
But it is a free market value to choose to perform under authoratarian discipline -- something that more doctrinaire libertarians sometimes forget.
Not having served, I could be wrong. But I have the impression that initiative is also a virtue that the military approves of -- one that is more prevalent in a free market society. I'd also suggest that courage and discipline (at least of the self-discipline sort) are as much free market as authoritarian virtues.
It is no distinction at all.
Armaments are not produced by Mom and Pop firms. They are produced by firms so hooked into the military-industrial complexes of their country that they might as well be state run. If an enormous amount of your business is done with the government you get this revolving door where the politicians and generals you dealt with become your future execs. And do you think they are allowed to fail ? Do you think the government would sit back and watch its defense contractors go out of business ? There are consolidations and bailouts to make sure they are viable and the capacity to produce advanced aircraft is not compromised. The rivalry between Boeing and the AirBus involves the governments of America and the EU. Big defense contractors really aren't "private industry". McDonald Douglas has more in common with Dassault or BAE or the Sukhoi Design Bureau than it does with WalMart.
And the 'initiative' of a mercenary if primarily focused on saving his butt, not getting the mission accomplished.