UAB fans can wish for it all they want. The first thing they need is the cooperation of UA officials and that's not happening.
Let's just say the UA administration is keenly aware of what the reaction would be within the fan base (and the donor base) if they were to ever schedule UAB. This isn't comparable to Alabama playing in Auburn; those were two schools in the same conference with one of them (AU) having a logically legitimate complaint at the time because Alabama was playing half or more of its home games in Legion Field every year.
UAB has virtually no fan support outside of metro Birmingham, plays in C-USA, is one of the schools immediately in danger if the NCAA ever passes a minimum attendance requirement and loses money like a man trying to restore a 1970s Jaguar pulled from a junkyard. A UA-UAB matchup benefits Alabama not one bit, but would benefit UAB a great deal.
As someone above noted, it's not UA's job to provide relevance for other in-state programs. Auburn fans can ignore the facts all they like but if Alabama hadn't begun playing them, Auburn would never have been a relevant school. And that was a mistake for Alabama -- one that I doubt UA makes again.
Also, I'm squarely in the camp that says UAB football should be disbanded or at least busted back down to Div-II or lower. The state of Alabama is not rich and UAB football is a terrible financial drain on an already stretched-thin system.
