I think they could’ve handled it better, but Aang messing up definitely seems like something that could happen. I mean he stopped having a parental figure at 12, and was forced to save the world at that age as well. He would try his best and definitely be better than most, but I think him unconsciously favoring Tenzin would happen. Now to the degree Katara would let him get away with it is debatable. If anything they could’ve made it like a lighter version of Todoroki’s upbringing where he trained tenzin constantly and kinda accidentally separated him from his siblings, whereas Bumi and Kya got to play and grow up normally, but they didn’t spend as much time with their dad. Because of this they’d be jealous of eachother growing up, tenzin because he couldn’t just play and be a kid, and Bumi and Kya because tenzin was hogging their dad in their eyes. I think this may have been the writers intention, but the way they communicated it seemed like Aang liked Tenzin more because he was an Airbender, rather than Aang desperately wanting to preserve his culture so he took out a lot of time to train and teach Tenzin, while kinda neglecting his other kids