I've read that the original cut of the film was quite different as well. I mentioned this to Gene Kelly's widow Patricia Ward Kelly on her FB fan site and she said she planned to cover that in her forthcoming biography of Gene.
Apparently they changed just about every musical number with the exception of "Nina" and the "Pirate Ballet."
"Mack the Black" was originally slated to open the film as Manuela (Judy) sings it to the girls on the veranda. I don't know if this was the notorious Kay Thompson arrangement or if it was re-arranged for filming, but you can still see a snippet of this in the original trailer. There's a suspicious cut in the final film where the song should have gone.
They moved "Mack" to where it currently is because the original song for THAT scene, "Voodoo" (a song and dance duet with Gene Kelly) had to go. They staged and shot it twice before finally giving up on it and if the recording that still exists on the soundtrack is any indication, it's no wonder! They needed more Judy in the movie so they gave her a solo spot which worked out as it should as it's been reported Gene Kelly had a broken ankle during re-shoots and couldn't really get around without pain. He had to sit on the sidelines to cover it.
Freed cut Judy's solo, "Love of My Life," from earlier in the film (we still hear the reprise right before the end) which Manuela originally sang right before Kelly "invades" the town. In the final cut, she pulls a hatbox out of her closet that contains the limp, ruined hat she wore when she first met "Mack" but that originally led into her singing the solo
"You Can Do No Wrong" was originally in the same scene that it currently appears but was placed a little earlier and was staged as an ironic, comic song in which Manuela mocks "Mack." Realizing they didn't have a "love song" between the two anywhere in the picture, they decided to re-shoot it as a traditional love song that is not ironic. Again, because of Kelly's broken ankle in the re-shoots, it's convenient that she sings it while cradling him in her arms since he couldn't really walk around that well.
I don't THINK Judy was originally supposed to join Kelly for "Be a Clown" but the decision to add her was obviously made during the original shoot because Gene Kelly seems hale and hearty throughout!