The ride stands on its own well enough that those who haven't seen the movie can get enjoyment out of it. But that doesn't mean it's "completely separate" from the movie. The songs and the characters all come from the animated portions of SOTS. The general plotline of Splash mountain is a loose adaptation of the Tar Baby story from SOTS (though the baby is replaced with a beehive).
I do think the Magic Kingdom version feels less directly inspired by SOTS than the Disneyland version. The Disneyland queue for the ride plays instrumental versions of the songs "Song of the South," "That's What Uncle Remus Said," "Sooner or Later" "Let the Rain Pour Down" and "All I Want" — which all come from the live-action portions of the movie. The Disneyland version queue also includes quotes from Uncle Remus and his fireplace. In fact, the entire namesake of Critter Country in Disneyland directly comes from the Remus quote "Where the critters are closer to the folks and the folks are closer to the critters."
The Magic Kingdom version has less overt easter eggs, but the story, characters and songs still come from the movie.