Well, first off, you really didn't answer the question about whether music is meant to be heard, not read (just as a play is meant to be seen, not read).
As far as writing lyrics and the music at the same time, of course they overlap...but at some point you either have to put lyrics to the music or music to the lyrics. So, which is it? It sounds like you prefer putting music to lyrics.
And of course the individual types of art connect, but that doesn't make them the same. Lyric writing is not the exact same thing as poetry in its more traditional sense, and I think sometimes people get in trouble thinking that it is.