Aside from the danger of them doing damage to your home, them chewing something toxic, them chewing through walls, electrical cords, etc etc etc, not to mention the potential for them to get hurt.. no, no other reason to keep the cage closed while you're away.
Sorry, was that my sarcasm coming out? Birds are like toddlers and a GCC is NO EXCEPTION! They are little, it is much easier for them to get hurt, not to mention meet an untimely demise. While my Loki is left "out" much of the day, anytime I am going to be gone from the house (other than outside for the dog or checking the mail), he is put away in his cage. Door closed. Period. My husband has accidentally left Loki out and we came home to find that Loki had chewed through his iPhone charger. PLUGGED IN! Loki could have easily been electrocuted. We could be very diligent, but we're never going to be perfect at making sure all the "dangers are covered", so to speak. Safest place for a toddler that can't be completely supervised is in a "safe space" or a playpen. The cage for a bird is no different.