Although they don't state an exact date, there are several clues that the story is clearly set after the Civil War. Disney has also said as much. Those living on the plantation seem to be sharecroppers who can come and go as they please. While the movie suffers from racist stereotypes, I did not see a glorification of white domination. I guess you could say they were idealizing the plantation setting, but it's hard to separate this from the standard idealization of any setting that Disney applies to its films. And as I previously mentioned, a film like Gone With the Wind idealizes the plantation setting about 1000x more than Song of the South. This may sound like I'm defending the film, but I'm really not. There are definitely big issues.
For those who may not know, the story is about a young white boy befriending Uncle Remus. Uncle Remus teaches him life lessons through short animated stories interspersed throughout the live action. The boy's mom seems suspicious of Uncle Remus and doesn't like them hanging out together. The boy's father is away on some trip and the boy has been missing him and wants him to come home. At the end, the boy is injured and dying and instead of asking for his father, asks for Uncle Remus because of the connection they've made. Uncle Remus, in his tattered and dusty garb is welcomed into this fancy room to see the boy. I seem to remember a scene at the end where there's a close up of them holding hands, to represent some sort of racial unity.
If you look solely at this story, it seems like a sweet film about a budding friendship between two people. And like a story about how children can be blind to the racial differences that bother adults, like his mother. However, one article I read put it well - the film is guilty mostly because of what it does NOT depict. Like many Disney films, it leaves out anything from the time that would be deemed problematic or controversial. The problem is, the whole situation itself is problematic and controversial, and by not addressing any of those issues - like why Uncle Remus is still living on a plantation at which he was likely a former slave, why the boy's mom is suspicious of him, why he seems to be so poor and live in a shitty cabin compared to the mansion next door, etc. - the film seems like it's only trying to show the "good parts" of plantation life during the Reconstruction era. The whole exercise feels like a writer's room of white folks trying to show that they're not racist by including a positive black character who saves the day, while also trying to avoid touching on any of the consequences of racism and slavery that may make audiences uncomfortable.
I honestly don't see any way that Disney could have made a movie about this time period that wouldn't have suffered from these issues. Their brand of idealizing periods or events in history that in actuality were probably pretty terrible does not work for something that is still so recent and still greatly impacting us today. They should never have made a film with this setting.
With all of that said, I don't think they should keep it hidden away. I certainly don't think it should be marketed to children, but I also think it does not help conversations about race to pretend that this film doesn't exist. We can't know where we are and where we're going, if we don't understand where we've been.