I think you're missing the forest for the trees. It's one thing to store the lyrics in the file, but the entire UX around the use of those lyrics is pratically non-existent.

Storing the text is the easy part. Procuring files with lyrics baked in or adding lyrics to your pre-existing library is another. And so far, almost no music player supports them, their UI isn't necessarily optimized for displaying lyrics, and there is no widespread standard for their format.