Why not use "of" or "in" ?

The "words to" is just a fixed expression with any sort of lyrics. We don't use those other prepositions. I don't know why.

"Those are the lyrics to the song." "Those are the lyrics of the song." "Those are the song's lyrics." ...are all the same "Those are lyrics in the song" sounds more like you're referencing a few specific lines in the song, instead of the whole set of lyrics.

If anyone would make a grammatical mistake, it would be Fox News, but they’re correct here. You would say “the words to _” when describing the words/text of something, like a speech or song. “Of” could maybe be used as well, but not “in”.

I don't know if there's a technical reason or if this is just how it's said. "The words/lyrics to the song" and "The song's lyrics" are the best options to express this.

"Words to" and "lyrics to" are commonly used. You could also use of, but not in. Note that most people will use to. In would have a very limited use, like "Some of the words in the song are explicit." If you're talking about the lyrics or words as a whole you usually use "to".

"the words/lyrics to" is a usage by convention. I'm also interested in the history of the evolution

It's not weird. "to", "of" and "in" are all fine. They may have avoided using "in" because of the other "in". There's nothing wrong with having "in" twice, but it's a little bit awkward, ie "...the words *in* the Pledge *in* the Senate".