I was thinking of the entire civilian job positions and this also came to mind; "Does the Federation contract with Civilian Contractors to provide basic and even military jobs such as "Security Personnel guarding VIP's. When you look at the number of Starfleet security personnel, not too many are shown outside of the fleet, so I was wondering if the Red Shirts assigned to a Starship aren't really red shiirted Private Military Contractors, hired to provide the bulk of security officers on a starship? Or act as troops according to Federation Policy. Remember "Blackwater"??? they were hired by the US Government to do the things that the military was supposed to do, but with a smaller military and in this case a smaller Federation Naval Fleet, maybe the idea of the red shirts are well trained Red shirts, who just happen to buy the 'farm' in every episode of TOS...
I don't know why it should matter if the Federation ever existed, the question is a valid one, did the Earth Government prior to the formation of the Federation ever used PMC's or continue to use the Federal Government Civil Service system to perform the many muyriad of jobs that the United Earth Starfleet would eventually perform? Yes, you're also correct i nthe assumption that the Federation did not come into fruition till years later.
"Are you Starfleet or private security?" - Scotty to a uniformed Section 31 henchman in Into Darkness.
The Federation uses private military contractors to fight it's wars because Starfleet isn't a military.
If Starfleet's not a military, wouldn't they work together with government militaries? Either the Federation's military, or if it doesn't have a military, surely many of the nations within the Federation have militaries.
Considering there were private mining facilities on the Lunar colonies during the ENT era and the Federation was using holograms by VOY for dilithium mining operations, I’d say yes. MACO is under the United Earth government banner as a part of their military, and according to Captain Balthazar Edison/Krall in ST: Beyond, MACO merges with Starfleet. So it's possible that private security exist after the founding of the Federation.
Private companies certainly exist, and I have no quarrel with the idea that private security companies exist. It seems possible that Starfleet would work with private security companies on occasion. What I don't understand is why private (business-owned) militaries would exist within the Federation, or that the Federation would work with private militaries. A military is a different thing from a security company, after all.