it quite literally IS abt his culture and about him feeling like if he dies everything about the Air Nation would die with him
"However, we know from LoK that the only person in his family he shared Air Nomad culture with was...you guessed it, his one airbending child."
not true at all. ofc yes, he did spend more time with tenzin but look at i from his perspective. he knew that one day tenzin would have to teach his reincarnation, so he had to make sure the boy knew EVERYTHING he knew, not just bending, culture, customs, practices, etc.
not to mention tenzin was a more wilder child than his siblings.
aang had his hands full with tenzin. also keep in mind the age gap between tenzin and his older sibs. by the time he was a teen, both bumi and kya were already adults and moved out.
the only issue the other kids had was that aang spend most of his time with tenzin and was too busy with work NOT that he didnt love them or didnt like them.
and that's the tragedy of aang's character and his role of being the avatar. yangchen said herself: being the avatar means that his sole duty is first and foremost to the world, not to himself.
also mind you THIS is how they ALL looked at their family photo.
they all agreed to have had a happy childhood:
"It's that Aang never tried to share things with Katara, his culture, his grief, nothing."
saying aang never shared his culture/customs with his family/katara is complete baloney:
and here he;s sharing her his grief:
now i'll admit the Gaang's writing in TLOK and after ATLA is... iffy and definitely not perfect. we have so many unanswered questions. but with the little snippets of details we have gotten we can see that aang did share his culture with his family and that each of his kids did get to spend time with him and had their own unique relationship with him.
It's not just what Aang says to Katara and how dismissive it is, especially in the context of Katara dropping everything to help him without question, and then the moment she does question things, he immediately insults her like this.
The thing that drew Aang to Taga was his airbending, despite the red flags that appeared from the moment Aang met him that signaled that Taga was nothing like the peaceful monks he knew.
i really don't see how you watch the movie and thing he insulted her. he's clearly in a lot of grief and mourning and self-hatred/survivor's guilt and that's the thing. grief is messy and that's what this movie is trying to show: how aang's grief makes him act reckless and blind to all red flags and honestly its a very human approach to aang's character.
(and what he said IS true. Katara might be the last bender but she still has her culture and her people. Aang has neither. so ofc no one can ever fully understand what its like)
while yes there were red flags, aang was desperate for community. someone he could share his customs with and learn from and who would actually understand him and his culture (ex. the "Air nomads are formed from the mist. We walk the world for a moment in time. Then we return to the eternal wind" -> something that is tied to his culture, that he could talk to.)
thats the whole tragedy of his character
now ofc, does the movie have flaws, yes. (ex. sokka's characterization) but this movie is about Aang. about his unresolved grief. and how, now that he's an adult and his brain is developed, he's understanding the true weight of what it means to be the last. which also ties in to why he spent more time with tenzin: to make sure that his culture and everything he knows wouldn't die with him.
but that doesnt mean he didnt spend time and share his culture with his family ESPECIALLY considering how happy he is to share it with the acolytes and other people:
so if he's happy to share his culture with strangers, what makes you think he wouldnt do the same for his family??
(comic panels are from TLOK: Patterns in Time, TLOK: Kya and the Secret of the Sand, and ATLA: The Promise)