Not sure how it works in SA, but in the SBL the venues are mostly local government owned Rec Centres (with a couple of "community groups" thrown in, eg Lakeside Baptist.) (and the Hawks play at the State Basketball Centre.)
I imagine that's a similar story across the country for the minor leagues.
At the end of the day, if a local community wants a (better) venue its up to the Council to drive it, get the grants, etc.
If a team wants a/c, they should lobby the Local Council. Having the state Basketball association take over responsibility would be a costly exercise, and a political nightmare.
As for the Adelaide Arena, never been there, can't really comment. However I suspect that purpose-built stadiums are a costly mistake.
Perth Arena would never have been built (and never would have got their central piece of land) if it weren't for the government. Ironically one of the primary drivers was keeping The Hopman Cup, and it is now our number 1 musical and theatrical venue.
I don't know who owns the scoreboard (maybe the Cats) but even without the Cats extra cash, its a lot easier to find money for things like that when the rest of the venue is already taken care of.