That MAY be true. But it was far from universal. And there were many local congregations that were, for their day, progressive. Carter, as he tells in his Our Endangered Values, believed his more progressive worldview was in complete keeping with his Southern Baptists tradition.
I may be mistaken, but I believe that Huey Long, a great progressive of his time, was a Southern Baptist. Clinton was also a Southern Baptist. Southern Baptists most certainly have had their fair share of progressives. Those progressives may have been an aberration, but they were made possible because, until the 80s and 90s takeover of the Southern Baptist Convention by fundamentalist like Falwell, free thinking was allowed in that denomination. That doesn't put them in league with the Quakers or the UCC, but it doesn't entirely segregate them from progressives either.