The USAF moves things around - people and materials. Hauling cargo is effectively an exclusive role for USAF when compared to the aviation branches of Navy and Army.
When you ask about it being a private airline... there are some real disclaimers there. It is not a case where military people rely on the USAF to fly them places for leisure travel. Understand that there are planes of every shape/role/size/mission flying daily. In many cases, these are training missions and they can be empty in terms of cargo with the exception of the air crew flying the plane. In cases where they are flying a training mission between bases or locations, there may be seats available for military people (active or retired) to hop on as a Space-A traveler. In that case, it is a little like a private airline, but you generally cannot reliably count on it if you are on a strict timeline. Planes break and times are squishy. In some cases, there are predictable routes between bases, and there are some people who do enjoy Space-A travel - but your schedule must be flexible and you may have to pay for a commercial flight back if the "mission" changes and that plane that was flying back to Colorado is now headed to Europe on emergency orders because one country invaded another. It is common for retirees to leverage this system because they can go on a trip and potentially come back when convenient for the USAF. However, if you need to be at work on Monday morning.... Space-A travel is risky. I would venture to guess it is more risky than trying to fly as a commercial standby passenger as an airline employee family member. Nothing beats a paid airline ticket, but when you have alternative means of flying, you need to be flexible.
At USAFA, I used cargo plane flights in two ways - one as a member of drum and bugle. It is the only way that group travels. People strapped in next to music instrument cases. Most sports teams fly commercial and the big teams fly private. I also used cargo flights while a member of the USAFA Karate Team. I knew our schedule in advance. We could reserve full government airline tickets (refundable) and then as a cadet, I could call Peterson AFB and tell them we had a mission - fly 5 pax from PAFB to (pick-a-big-city) departing Friday morning and returning Sunday afternoon. That request went into the system and a reserve unit would answer the call - book the mission (cancel the commercial ticket), and we had a ride to a tournament that weekend. If no one picked up the mission, we would fly commercial with funds from the club budget. If a military ride was scheduled - that was $0 on the budget. We would get off the plane and the crew would get off with their golf clubs in hand. Occasionally, the plane would "break" and we have to stay over one more night while it got "fixed." NO ONE complained because it was an excused absence from school...
