That song is directly inspired by the Pepsi nonsense; the lyrics and backing video [1] have some direct quotes. Definitely the first thing that comes to mind each time a multi-billion company decides to change their logo and nobody likes it.
"Redesign Your Logo" was actually directly inspired by this exact brand redesign document, so your instinct is well-founded! Here's a transcript of the official Spirit Phone commentary track where Neil mentions it:
It immediately makes me think of the song "Redesign Your Logo" by Lemon Demon. Pure comedy.