To win in a Southern state around that era, Democrats needed two things:
1-A solid reliable core of Democratic votes in a large urban center that weren't nullified by suburban Republicans.
2-Still strong support from Dixiecrats in rural areas.
Georgia had both. South Carolina, Alabama and Mississippi lacked #1, (Birmingham doesn't go anywhere near as far as Atlanta does, especially because back then basically all its suburbs were Republican and Jefferson County was a swing county), Texas and Tennessee lacked the second half of #1.
Georgia wasn't really that unusual though, Democrats still held up well in Louisiana (which also had both), North Carolina was still a swing state, albeit in a different way than its a swing state today, and Democrats still controlled the state legislatures of Alabama, Mississippi and Tennessee up until 2010, later than Georgia in fact.
I don't even think they even necessarily needed a major city. Little Rock isn't that big, but Democrats were able to stay strong with a Dixiecrat coalition in Arkansas (the last Southern state to fully flip) until 2014.
