Doubtful. Not all music is the same, nor are the people who write it. Songwriting in it's more poetic forms can become as complicated as writing poetry itself. Writing lyrics gives you a lot of options, and with that said the rest of what you've said is validated, but the fact that Britney Spears and Iron Maiden write lyrics doesn't take away the fact that songwriting can be as difficult as poetry if that's what you're aiming for. Some people write their songs to be deep and full of meaning, some write to hang of one single hook line.
I don't know how you got the idea that "you are forced to tell a much simpler story, as there isn't space for the complexity" with restricted space, it forces the writer to make their points as deep as possible with the shortest amount of space, much in the same way that a single line of poetry can contain several different kinds of meaning within it.
You don't need a lot of space in order to establish meaning, some of the best poetry i've read, say from Coleridge or Browning, have written very short poems, but the content within the poems are enough to write long essays about them. Try "Meeting at Night" by Browning, for example. And that is the same case in songwriting, you've got even less space to flesh out your points, but that doesn't mean the piece can go without meaning.
