PG-13 tends to be very limiting in regards to the amount of blood, sexual content, and language a work can have. So why not go all out with a TV-MA or R rating? I think giving the Avatar Series the Samurai Jack season 5 treatement could open up new roads for the series to explore.
You mention that Sokka's sharp weapons never hit anyone, but I can imagine a scenerio taking place during Book 1, where the Gaang are fighting a platoon of Fire Nation soldiers, and one of them manages to corner a Katara fresh out of water. As he's about to burn her, Sokka comes rushing in. throws the soldier off his balance and starts bludgeoning him on the ground until he bleeds to death. Aang, believing all life was sacred, shockingly asks Sokka how he could take another life. Sokka replies that he wasn't going to lose another family member again and points out how Aang would put his own selfish desires over protecting his friends. Katara then comes to Aang to help him understand that sometimes you need to do whatever it takes to survive.
Another one would take place during the episode they meet Jet. Where they can't proceed on their journey, neither on foot or air because the Fire Nation has a heavy presence in the valley. The Gaang fails to stop Jet's plan, and he succeeds in wiping out the village, causing the Fire Nation to fall back and let the Gaang pass, but at the cost of dozens of innocent lives.
There'd have been blood, there'd have been the occasional cuss word (my bet is still on "bisonshit" being a common one), they'd have been allowed to more explicitly talk about death, and they'd have been allowed to make it more unequivocally explicit that Korra and Asami were falling for each other.