The Henry I found doesn't match well - he was alive in 1920 and 1930, so I'm starting to think Henry's a red herring. But the James born 1867 is a strong possibility. If he was alive in 1911-1912, fathered James with an unmarried Mary Alice Gardner, and died before they were ever married, she may have given him the name Joseph Brown or even Joseph Gardner Brown, but been unable to raise him.
In comes his Uncle Aleck, who raised him until he was old enough to take care of himself. I did find a 1930 Census with a Joe Brown in Richmond, Georgia who appears to be working on a construction project. It seems to fit. Again, speculating, but when the project ended, he probably returned home to Barnwell, married Susie, and fathered James Brown. James was born in 1933, Susie left him in 1937-38, and by 1940-ish, he picked up and moved to Augusta.
I did find a potential Mary Gardner on a 1910 Census, but I'm not terribly comfortable with her as she is listed as married in 1910, having 3 children, but the husband isn't listed anywhere. She's a possibility, but it would imply that Gardner is her married name and that either Joseph was born while James was with a married woman, or she was mistakenly listed as married when she was widowed, and again, perhaps he didn't live long enough or wasn't interested in either marriage or possibly even raising a child.
Lots of speculation. Wish I had good concrete answers.