See how Ron is using "government personnel" vs FAA personnel? He knows there are numerous facilities that are military approach controls, mostly Air Force and a few Navy (not sure about Marines). The Air Force operated Berlin Center during the Cold War, and I've been told the center over Nevada/Area 51 many years ago, don't know that as a fact though. During the controller strike/firing in '81 military controllers were sent to FAA facilities all over the country (myself Quonset Pt Tower), which included towers, approach controls, and centers. They did the job and kept the system operating on a limited basis until the FAA could hire and train replacement controllers.
Point is, if military controllers can show up and work these facilities (granted they had experience) then a private corporation could provide the same services as the FAA. There would have to be an overlap until the "new" company assumed full control but it could be done.