I was thinking about it the other day, and I think the intro is important.
The whole song is about coming undone, and letting yourself "come undone", so the contrast between the long, sustained organ notes at the beginning of the song to the fast-paced piano rhythms at the end is another way in which the lyrics are reflected through the music.
I think Vanessa intended it to be like this, and that's why I can appreciate how it fits into the song as a whole