Yes, if UAB can have a football team that is not a giant money drain, then more power to them. I'm not sure whether they can do that or not, but I am sure they cannot do it at the FBS level.
In the state of Alabama generally and in the University of Alabama system specifically, there is more demand for money than there is money to meet demand. Spending money for FBS-level UAB Blazer football, with "crowds" of a few thousand fans at best, is an ill-advised luxury. That Birmingham/Jefferson County has used tax dollars to buy tickets to the games and gave them away so that UAB can maintain FBS status is scandalous.
This may sort itself out naturally as the so-called Power 5 move on to another level....