Morgana and Kayle's conflict already was besides good vs. evil. Morgana was outcasted by her family because she choose to rebel against an authoritarian society which Kayle idolizes, she herself embodies this dictator like attitude in her Aetherwing skin, where by this point the timeline has eradicated Morgana and the rebels. Individualism vs Collectivism.
A tidbit from Morgana's Lore -
One side proclaimed themselves as beings of perfect order and justice, fighting to unite the world under their law and strong central governance. Those that fought against them saw their kin as tyrants, creatures incapable of seeing the larger view, who would sacrifice individuality and freedom for the illusion of efficiency and safety.
...
Kayle struck the first blow by disowning any filial connection when Morgana refused to join her cause.
Now Kayle -
For ten thousand years, Kayle fought tirelessly for her people, wielding her flaming sword forged before time itself.... Though a beautiful, striking creature, Kayle, now as then, avoids showing her face; war has taken a terrible toll upon her spirit. In her quest for victory, she sometimes would try to lift the wicked up from their morass of evil, but more than often she instead purged those she herself deemed beyond redemption. To Kayle, justice can so often be an ugly thing.
...
and bring her own brand of justice to the League of Legends.
Morgana and Kayle share a very complex relationship that cannot be mend due to differing ideals.
Also addressing Demacian society, Demacia is a city state that stands for justice and truth at all costs. AT. ALL. COSTS.
Lux was a former spy, Fiora outcasted because of her father's lies, and I'm sure other evidence would prove that Demacia holds it's ideas too high. The Lore right now is pretty cool, but change happens, all around us - lets just accept it.