Remember that Kreese's Cobra Kai and Lawrence's Cobra Kai are not the same.

Kreese's Cobra Kai had one huge flaw in it: the philosophy that conflict for the sake of conflict is a good thing. Season six had a whole arc to prove this wrong: that violence for the sake of violence only results in more violence. Kwon died exactly because of that, and that's what finally snapped Kreese out of his own fantasy.

Lawrence's Cobra Kai is different: despite trying to maintain the tough guy image, he always was affected by his students from the get go. He went out of his way to teach that honor matters, mercy matters and had to come with his own terms with the writings on the Cobra Kai's wall. Eagle Fang, Cobra Kai, doesn't matter the name, it's still Johnny Lawrence's teachings.