You are half right, according to the song writer Robert Lopez the original movie structure and plan just had the placeholder title “Elsa’s Badass Song” and he was told to go write it, and then came up with let it go.

But it was still very early in the process and it was the first song to be written that actually made it into the movie. The extended soundtrack includes quite a few of the early songs that didn’t make it into the movie where they were still trying to work out what the style of music was going to be and who the characters are, so the music and story were definitely co-developed together.