I will die on this hill, none of the Avatars said to kill Ozai.
Aang, and you, just read it that way, because they were being more philosophical and less practical than might be hoped.
Roku: In my life, I tried to be disciplined and show restraint, but it backfired when Fire Lord Sozin took advantage of my restraint and mercy. If I had been more decisive and acted sooner, I could have stopped Sozin and stopped the war before it started. I offer you this wisdom, Aang: You must be decisive.
Kyoshi: In my day, Chin the Conqueror threatened to throw the world out of balance. I stopped him. And the world entered a great era of peace. ... I would have done whatever it took to stop Chin. I offer you this wisdom, Aang: Only justice will bring peace.
Kuruk: When I was young, I was always a go-with-the-flow kind of Avatar. People seemed to work out their own problems, and there was peace and good times in the world. But then, I lost the woman I loved to Koh, the Face Stealer. It was my fault. If I had been more attentive and more active, I could've saved her. Aang, you must actively shape your own destiny and the destiny of the world.
Yangchen: Many great and wise Air Nomads have detached themselves and achieved spiritual enlightenment, but the Avatar can never do it. Because your sole duty is to the world. Here is my wisdom for you: Selfless duty calls you to sacrifice your own spiritual needs, and do whatever it takes to protect the world.
What the Avatars actually told Aang was, "you cannot hide from this, you cannot run away and pretend this isn't happening, you can't stall and hope someone else will fix it for you."
He was decisive, he brought justice (not vengeance!), he shaped the destiny of the world (like, a lot) and did what it took to protect it!
Maybe they couldn't think of a way to do it that wasn't murder, we're not in their heads, but they left the option open for Aang to come up with another way. As long as you save the world, you're good. But you have to save the world.
I feel like I've made this post before, but here it is again.
