My idea for LoK is that each nation developed their own cultivation style in the same way they each bend one element. Avatar always stood out in that each nation has a strong sense of identity, but the thing is Zagreus introduced cultivation and is a 'god of foreigners' so the greek style of cultivation is popular and prestigious in places like Republic City where people are more independent-minded and individualistic. The modern myth of the self-made man married the romance of the lonely rising star.
Zuko is the Fire Lord and the hundred years of war coupled with how Zagreus conquered it led them to a public virtue-inclined style of cultivation. Zuko himself has honor as his capital virtue so things are more tense than in canon. As you said, Zuko never ingratiated himself with the Avatar gang.
The Earth Kingdom broke up due to Zagreus interference so now it is comprised by City States bickering with each other like greek polis defiantly fighting for independence despite the constant rise of would-be 'Great Unifiers' trying to conquer their neighbors.
Sokka is the chieftain of North Pole and the position is affecting him, every tyrant was once the greatest in a parade of one hundred heroes.
In the middle of it all, Azula is Avatar Ra's Al Ghul, the bane of rulers. While in canon she dedicated her life to making Zuko a strong Fire Lord by opposing him, in this version she took on a more global approach. Zagreus's conquest of Ba'Sin Se and Fire Capital left a strong impression and all wannabe rulers of men know to dread the visit of crows.