It's entirely possible. Up until a few years ago, there was a flight path in the USAF known as "career trainer", akin to professional CFI in the AF. You spent your career in Training Command, and that served as your AF job. Most of the people that got this job were those that were non-deployable for one reason or another, and many people also requested this duty. It's not around much anymore, but my IP in pilot training was an O-5 that was a career trainer following 1 tour in the B-52.
Aloft:
The C-21, C-9, and other "staff" type airplanes aren't considered Major Weapons Systems, unlike the C-5/C-141, etc. When one gets an assignment to OSA (Operational Support Airlift) after pilot training, it's assigned by personnel center as your job, plus a follow-on.......for example "C-21 to Randolph with a strategic airlift follow-on". You only fly OSA for 2 years, then move on to your follow-on airframe in whatever you're assigned (strat airlift, tactical airlift, tanker, etc). That's why your friend couldn't stay in C-21s.