Akira is an '80s movie. The scenes in FFVII were inspired by it, GITS, and other cyberpunk/New Age.
And FFVII does it better.
What inspired what has nothing to do with quality on an individual level. Akira just does not stand up anymore, IMO. GITS is decent, but not mind-blowing by any stretch. It certainly does not sum up the series.
If someone has gotten their idea of anime from Akira, GITS, and Dragon Ball Z, it's gonna be an understanding so narrow so as to be useless.
I'd suggest someone on a course from Vandread (introduces you to the anime style of throwing you into the middle of the story without explaining a goddamn thing), through Cowboy Bebop and into Bleach (because Bleach will be running in the background forever with 335 episodes and counting), then add on FLCL and Gurran Lagann and maybe a Gundam series.
Really basic but covers a spectrum.
Akira and GITS can be picked up later as filler. There's no hurry in either.
I suggest Akira, Cowboy Bebop, Memories, and GITS when probing for Anime tastes. NEVER DBZ, Bleach, Naruto, Spirited Away, Evangelion, etc.
Probing != simply suggesting something to watch. If you have someone impressionable who will follow your follow-up suggestions, Akira would be useful to get an idea as to their tastes. I would not suggest it to someone whose tastes were more set, though.
As for Dragon Ball Z, Naruto, and Neon Genesis Evangelion (never seen Spirited away), I agree that those aren't good intros. I dropped Dragon Ball Z at the fight with Frieza. (couldn't even finish the fight.) I tried Naruto and dropped it after wincing my way through 6 episodes. Neon Genesis Evangelion is ok, but it's probably not going to engender a great liking for anime.
Bleach is not those, though. Just because /a/ shows tsun-tsun for all the Big Three doesn't mean they feel they are equal. They certainly have a special place for Rukia.