SoTS was a bit of a box office disappointment in 1946. Its financial success was due to multiple re-releases. In 1946 it wasn't the embarrassing flop that movies like Fantasia were, but it fell below Disney's expectations, and the controversy and protests didn't help either. The stereotypes and romanticized plantation setting featured in the movie were commonplace in the 1930s, but by 1946, critics were already finding them to be outdated and offensive. The 1956 re-release was less controversial, but it also made less money. It sorta flew under the radar. The movie was not re-released in the 60s due to the tensions associated with the Civil rights movement. However, segments of the movie did air on TV, and the Brer characters had a large presence in Disney parks.
Song of the South's real popularity came in the 1970s. While the movie wasn't a critical or commercial hit in 1946, there was an entire generation of children who grew up with the Little Golden Books, record players, comic strips, toys and other associated Song of the South merchandise. Plus, Zip-A-Dee-Dooh Dah had become a Disney anthem. So when the movie was re-released in 1972, it saw a surge in popularity due to the adults who had a nostalgia for SotS and its associated merchandise when they were children. The 1972 re-release was the most profitable re-release of a Disney movie at that time. This release also didn't see much backlash, and the reviews were much more positive — due to a combination of nostalgia and a reevaluation of the film after the passing of Walt Disney. The 1972 re-release was so popular that Disney re-released Song of the South again in 1973 as a double feature with the Aristocats, in 1980 for the 100th anniversary of the Joel Chandler Harris Stories and in 1986 for the movie's 40th anniversary. There was mild criticism of the two 1980s re-releases from some Black writers, but because the 1980s was a pretty conservative decade the criticisms against the film were overshadowed by those who were fans of the film. The 80s were a very anti-PC time so a movie like SotS wasn't as offensive to the general public as it would be in a more PC era like the 90s. And as the movie got older, there was a growing revisionist narrative that the movie was just "of its time" and that no one really found it offensive in the 1940s.
I do wonder what the reputation of the movie would have been in the ensuing decades if Disney had simply released the film on VHS in 1987 after it's final theatrical release and kept it available for home viewing afterward. On one hand, more people would have been aware of Splash Mountains ties to a movie with racist stereotypes. ON the other hand, the availability of Song of the South might have made it a less notorious and controversial film. It may have actually become more "forgotten" as it would have no longer been this mysterious object of obsession by both fans and foes of the film. I believe if SotS had a VHS and DVD release it would have ended up like Swiss Family Robinson, Davy Crockett and Pollyanna — a movie fondly remembered by Baby Boomers and Gen X and largely ignored and unseen by millennials and Gen Z.