Igor - that's a great 'Emperor's New Clothes' question. The conventional wisdom, which I've accepted - and may be true - is that the term 'aikido' was selected as a catch all for any and all koryu grappling type arts (torite, jujutsu, kenpo, kogusoku, etc). The idea being that whatever name was chosen, those referred to by a different name (yawara as opposed to koshi-no-mawari, for example) would choose to be insulted.
Hirai sensei had experience with Takeuchi-ryu as well as a number of other arts, and was actually "During World War II, Hirai was the head of the jujutsu department of the Japanese Army's military police school and was instrumental in developing a new arrest technique employed by the military police." So, somehow, he was, even then regarded as far more than Ueshiba's student.
But it's a fair question if there was actually a 'category' of aikido that subsumed all these arts. I just did a cursory search in Japanese, and it was actually a little hard to find references that weren't related to sites associated with Hirai Minoru. I wonder if it was Hirai's own 'baby' within the Butokukai - and never really went anywhere in terms of organization or enrollment. Remember that Hirai, although the general manager of the Kobukan, claimed/regarded himself as a younger peer of Ueshiba, not a student, and that what he was doing was a form of parallel evolution. I wonder if, in fact, the only 'members' of the Aikido-bu were Ueshiba's aikibudo and Hirai's Kogado.