Guys, guys... Just a quick thought: what if every Avatar focuses on improving one part of the problem? And it's all linked to their first elements. It's just a rant, treat it like that.
Like Aang, an airbender. Airbenders are the ones most focused on spirit realms, however Aang's biggest achievements are building the Republic city and ending the war, both kinda materialistic goals (like one would expect from Earth bender, since Earthbenders are more about materialistic things, rocks and so on, and so on.)
That goes along with what (I think, I, sadly haven't read the books yet) avatar Yangchen also did. She focused on humans (and sided with them most of the time, leaving spirits angry) and ending wars as well as mentioning peace. Could be seen as materialistic.
And now with the new movie: as I see it, avatar Sonam's most prominent tasks were to establish peace. And how did she do that? By making a staff and creating new air benders. And when that heavily backfired, she sent the airbenders to the mountains. So she left her handprints on the relocation (building a new space for air nomads - materialistic) and creating
So I'm saying: materialistic legacy (even if it's kinda funny, that air NOMADS are the one in this category. But that just proves that avatars are not there as a representative of just one nation but of all the people. Or something).
As far as I know, both avatars from the Fire nation mostly focused (or what we know about their time) about inner politics. Szeto is said to be the first one to lava bend and to focus on internal stability and politics. Avatar Roku is mostly known (even if he did a lot of things!) in the Fire nation court as well (and shown as he is trying to stop his friend from the war), which could also be said to be inner politics and ensuring stability.
It could be: political engagement legacy. Or something.
And what about avatar Kyoshi? Well, she did a lot of things but what lasted from her time are Kyoshi warriors and Dai Li. Both are human groups that were established by her and survived centuries doing the same (or almost the same) thing as in the beginning.
(I also want to note, that she had a problem with who the true avatar is. And something (not really but) a little similar is happening in the new avatar: seven heavens... Just a note.)
Then there is Korra, a waterbender first and foremost. And what does she focus on? The spirit world (bcs she kinda has to, but still). That should be the airbenders expertise, right?
And avatar Gun (if I remember correctly) lost his friend in a tsunami and vanished for some time. When he reappeared, he was cynical and deemed the humanity undeserving of the constant saving. So he sided with the spirits, one could say... But we don't really know that much.
And that's the same with Avatar Kuruk with his heavy focus on the spirit world.
Therefore: spirit world legacy.
Let me put it in less words:
Fire - inner politics focus
Earth - organisations legacy (because there's really no one else to compare Kyoshi to :C)
Water - spirit world focus.
And now, let me link it to the elements. Air is all about freedom, movement and the spiritual world in my opinion (I kinda need to rewatch AATLA...). But they leave behind buildings, structures and so on to PRESERVE the culture or create a new one. A little bit of paradox there.
On the other hand, water is about healing (as I see it, again...) and the known avatars focus on healing the spiritual world (or so is my interpretation).
And that leaves me with this idea:
Water is the opposite of air.
Sure, it could be mainly because water focuses on the issues air causes right before but shhhh. Let me continue.
Air, in its core, is all about redirecting energy, about taking a stone and with a wind making its path different. It's fluid and all about flow.
And so is water, flow and fluidness. But the philosophical core is about containing the blow of water, embracing it and then changing the ENTIRE flow. Not redirecting it, but standing strong and changing it as it goes.
So air avatars instead of completely changing the present instead try to build something, preserve it. Just redirect it a little bit. But water avatars try to change the world by changing the spiritual energy. You can change humans, but they will forget the lessons in a few generations. But changing the spiritual world? Spirits will remember it forever. They absorb the blow that the spiritual world is in disorder, and completely shift the flow of how it goes.
But what about fire and earth?
Let's start with earth, because it goes right after water. It's very similar to water with the philosophy of enduring the blow. But instead of redirecting it, earth creates another one to destroy the previous one. No redirections, just creation.
It makes sense, for Kyoshi to create a new group to deal with the issues in the human world. She created a permanent human organisation, that will withstand blows and endure for centuries. And she had to do it, because the world needed human connections after so much spirit work.
And now, fire. Fire is similar to air and earth as well. Fire doesn't absorb blows, no. On this front, it's similar to air. It uses speed. And then they strike. That's the similarity with earth, they want to overwhelm the enemy with the attack. The strike.
So they impose new rules, choose how the kingdoms are going to be ruled and burn the corrupt structures off (that's where Avatar Roku failed sadly). After the cycle of earth avatars trying to do things outside of the inner politics of kingdoms and such, the fire avatars need to focus on the inner structures and change it from within, not from the outside. They can't absorb the blow, they can just dodge (like air) and then attack directly.
And that leads us back to air, air that needs to focus on preserving the culture from being wiped out, so it can outlast political chaos. Culture and traditions are neglected by the fire avatars mostly. There is almost always some war for the air avatar to clean up (due to inner politics not being stable as fire is not stable also? Maybe.) and restore the peace.
OFC, there are a lot of other styles and so on, but every element can have styles inspired by others and so on. There is no element that can't benefit from learning about others. There's a lot to talk about each and every element and to compare them and so on and on... But I would like to stop for now.
Alas, I believe the new avatar: seven heavens will be about creating long lasting human organisation and structures. But who knows, maybe not (kinda hard thing for two kids to do).
Also - I need to apologize for my English and a lot of stereotypes about the nations and past avatars. I'm a fan, but not really a huge one. This is just something that's been on my mind for a moment and I wanted to know others' opinions.