I never did magazine distribution as a kid. But as an adult, if I did not care for the subject matter, I highlighted the other magazine (Awake! vs. Watchtower) or some other article within. I typically focused on positive nature stuff from Awake!
But the term, "Those awkward magazines" really makes me think of articles that Watchtower corporation printed and now the articles are glaring reminders of their lies, contradictions, changes.