I unplugged my VCR over a decade ago, so I'm not going to try and go find it in the garage and rewatch this movie.
But as I remember it, Uncle Remus lived in his own home and had the freedom to socialize with whomever he wanted and do what he wanted. He also had the ability to wander the countryside of his own free will. He was clearly not a slave. And the movie was also clearly not taking place during wartime. Thus, we are left with the only option that Uncle Remus is a free man living in "old times" but obviously during Reconstruction.
And in other sequences, particularly when the other workers are singing and working together, it could very easily appear to be slaves in slave quarters.
Surely you can see how not explicitly dating the film-which as I mentioned earlier would have taken almost zero effort on their part-has not helped the movie's reputation or its audience's understanding of it. Expecting audiences to take that one sign as proof that it's Reconstruction, when Reconstruction is often not covered in depth, clearly did not work out to their benefit.
Perhaps you received extensive coverage of Reconstruction back in the day. That is no longer the modern experience, and I'm sure it hasn't been the experience at most schools in a long time. Not saying that such a change is for the better, but merely stating a fact.
Here is a link to the film, in the event you or anyone else would like to view the film or particular sequences: https://archive.org/details/SongoftheSouth1080pRestoration