Except even the article acknowledges that the stick's just doing the job that fins normally would do on other rockets. I know from first-hand (well, 1.5 hands; my stepdad was the one building and launching, and I was just there to watch and occasionally help out) that model rockets are capable of putting some decently-heavy payloads quite a ways into the air without a 20-30 foot long stick, and while model rockets are typically designed to be a lot sturdier than bottle rockets, they're also typically designed to be reused whereas bottle rockets are not (if such a requirement didn't exist, then they could be built to disintegrate and explode in the air, and be dirt cheap).
While I can't seem to find any specific rules from the FAA, Tripoli, or NAR specifically prohibiting the launch of a rocket that deliberately explodes in mid-air, my impression from my recollections of talking to expert rocketeers at launches (and asking them the sorts of questions a 10-year-old might reasonably ask, like "can I launch a rocket from an RC plane?" or "Can I launch a rocket from another rocket?") is that this would be thoroughly frowned upon my launch officials and/or regulatory bodies, and if the rocket exceeds the specifications for "Class 1" it'd require prior FAA authorization (and I can't imagine "Hey I'm gonna strap an H motor to this explosive ball and hope for the best" to go well on that front).
While I can't seem to find any specific rules from the FAA, Tripoli, or NAR specifically prohibiting the launch of a rocket that deliberately explodes in mid-air, my impression from my recollections of talking to expert rocketeers at launches (and asking them the sorts of questions a 10-year-old might reasonably ask, like "can I launch a rocket from an RC plane?" or "Can I launch a rocket from another rocket?") is that this would be thoroughly frowned upon my launch officials and/or regulatory bodies, and if the rocket exceeds the specifications for "Class 1" it'd require prior FAA authorization (and I can't imagine "Hey I'm gonna strap an H motor to this explosive ball and hope for the best" to go well on that front).