Paul McCartney wrote “Let It Be” for Aretha Franklin. He offered the song to her with the right of first release. Aretha recorded it, but then she hold up the release. So she only released it after the Beatles had a smash hit with it (according to Jerry Wexler and David Ritz in the liner notes to the CD “Aretha Franklin – The Queen of Soul”, 2007).
I think this story makes sense. “Let It Be” has a strong spiritual undertone and I imagine that Paul McCartney thought it a fitting song for Aretha Franklin who originated from gospel music.
There’s also a rare version of “Let It Be” by Tennessee Ernie Ford (thanks to Dusty Fingers for pointing this out), who during the later part of his career recorded many spiritual albums.