Yes, Bryant-Denny Stadium was build in the late 20s. However, Alabama has a history of playing games all over the state, including Birmingham, Mobile, and Montgomery. Alabama used to play teams like LSU in Mobile from time to time. But Birmingham was the site of most of Alabama's big home games - Tennessee, Auburn, USC, Penn State, etc. The Iron Bowl with Auburn was always played at Legion Field in Birmingham, and was split 50/50 as far as ticket sales go.
Birmingham was the center of the state's football recruiting, and Coach Bryant liked playing big games there for recruiting and tv purposes. Legion Field was bigger and in better shape than Bryant-Denny. However, 2 things happened. First, Alabama began investing in the expansion and enhancement of Bryant-Denny and by 1998, it had a higher capacity than Legion Field. Also, NCAA legislation in the mid 90s curtailed hosting recruits at off-campus stadiums. By 1999 and 2000, the Tennessee and Auburn games had moved to Bryant-Denny, and in 2003, Alabama played its last non-conference home game there, vs. South Florida.