I was curious why the Hogle Zoo didn't have flamingos, and this answers my question. The aviary and the zoo complement each other quite well as the zoo has a relatively small bird collection and no South American exhibits (other than some species of reptiles and monkeys in the primate and small animal complexes).
I kind of feel like if the zoo wanted to have a bird collection they could easily do so without overlapping the aviary's bird collection. Tracy Aviary focuses on North & South American species, whereas Hogle focuses on Asian and African species. I think because the aviary has a strong bird collection the zoo has put birds on the back-burner, having only parrots, raptors, and a "generic" aviary of smaller birds. The contracted bird show at the zoo has lots of excellent species but those birds do not stay at the zoo in the off-season, and they are not on display other than during the show. Not having standout bird exhibits or species as part of the permanent zoo collection is a missed opportunity in my opinion.