Most songs that are worth a damn have something inherent in them that sticks with you in some way. You could call that a hook (not always comprised of notes; it can be the beat also, see White Stripes' "My Doorbell"), or a pretty melody, whatever. These are elements of "classic pop songwriting," albeit perhaps not a clear picture of said. If you're not striving to include these somewhere, you're probably not writing decent music.
You'll also find that some acts are a whole lot more responsible with that "perfect pop song" ambition than others, and it fits them naturally. U2 is not one of those bands. On the other hand, I can't remember the last time Dan Auerbach wrote a song I actively disliked, and that's because his songs are reliably comprised of strong hooks/melodies. Every time. And the Black Keys actually seem to be implementing them more prominently these days, which seems to please most people. Their music, at its worst, involves competent blues rock cycling through predictable blues progressions. Yawn. At its best, kinetic, aggressive performances support a damn fine tune. That's the way to go.
If nothing else, I at least believe that the act of sitting down and writing something worth remembering should always precede the recording process. 9 times out of 10, you'll find an emphasis on songwriting will improve your songs, not weaken them.
This is all, of course, in the context of popular music. Not top 40, necessarily, but certainly excluding genres that would never, ever rely on "classic pop songwriting" to begin with.
