Air Force Space Command wasn't established until 1982, Star Trek was created in 1964. What other unit or command group uses or has used something similar?
Quite a few, actually. The emblem of the Arnold Engineering Development Center (founded in 1950) is probably the most interesting, but so are those of the 36th Wing (1952), 138th Fighter Wing (1957), 75th Air Base Wing (1942), 412th Test Wing (1957), and the 30th Space Wing (1964)
And those are just ones that predate Star Trek. More modern examples:
The Starfleet Command logo was not shown on TV until 1996 as part of the episode "Rules of Engagement" in the 4th season of DS9. It never appeared in the original series nor TNG.
http://www.ex-astris-scientia.org/inconsistencies/sf_command_emblem.htm
You could maybe argue it looks like it was inspired by the TNG communicator badge, but didn't go on the air until 1987.
So they really had no excuse this time.