The benefits are (were) the support structure, especially during times of plenty when surplus T-34s and T-41s could be the backbone of the club, on base facilities, including avgas availability (which is gone with the wind, obviously).
But you're right, the golden goose was strangled by the typically Air Force answer to non-existent problems, such as the hamfisted application of an AFI11-2AeroClubV1,2,3 template that turns a sunny day in search of the $100 burger into, well...more queep than it's worth.
That the Air Force doesn't make support (real, risk managed, managment-managed approach) to supporting Aero Clubs is a shame. It should be a core cultural priority; we're the fucking AIR Force, for chrissakes! AFB's should have an Aero Club, Navy bases should have a sailing club, Army bases should have an MWR shooting range and backcountry hiking/camping.