I wanna make one thing clear- just cuz I believe that it makes far more sense to kill Ozai, doesn't mean I blame Aang for not doing it or in any way question that it was Aang's right to make the final decision.
Execution is a serious thing, an action that is only to be carried out for one of two main reasons: Either the individual is too dangerous to be locked up, or else the criminal's crimes disqualify him to live off of the amenities of the people's government.
Such an act can only be implemented by the people's authority; it is their society being protected, their government and their freedoms and thier safety. If an individual threatens their lives, they have the right to lock up, or in certain instances kill the individual for the good of the people.
In our systems of society, the people elect the government to make such choices on their behalf.
Aang was not elected by the people.
However, Aang is the avatar, which according to the show means it is somehow determined by a higher power that his role is to keep peace in the world by whatever means are neccesary, including executing dangerous individuals. So, in essence, it was Aang's right, when acting for world peace, to kill whomever he had to. It is also his responsibility to decide if criminals should be punished in another manner, such as locking them up.
There are reasons why we don't execute everyone who deserves it: they may not be guilty; there is always that chance. Also, if you believe in an afterlife(which I personally do), then you realize that if you send a killer to be judged before they have a chance to repent, you are dooming a soul to hell. We would rather give them a chance to redeem themselves, atleast partially, and atone for their sins. This is partially why heroes like Aang try to avoid killing, and why Aang felt refrain from executing the firelord.
However, many individuals will never repent. They will never atone for thier wrongdoings, and will cling unapologetically to thier past in the firm belief that they were right and everyone else was wrong. I believe that Ozai is one of these individuals- he scorned the mercy aang showed him, and called him "weak". In prison, he mocked Zuko for coming to see him. He was a phsychopath who couldn't be reasoned with.
Such individuals are taxing on the system- they require food, water, shelter, and security. They are very expensive wastes of resources and money. They are allowed to live in prison for the rest of their lives in order to protect the people whilst giving them a chance to repent, which they will never do.
In my opinion, if they will never make up for thier crimes, then serious criminals are safer to society when they are killed. Otherwise they still pose a sligh danger to society and waste the resources which that society has generously provided.
Since the firelord will probably never relenquish any atonement, he is a waste of resources and a continual danger to the world(he may not be able to bend anymore, but he must certainly still have supporters). In my opinion, he should have been eliminated.
Nevertheless, Aang had the authority to make the call in behalf of the people, and I respect the people's decision made through him.
