There are good reasons to not leaving Ozai alive that a child wouldn't factor in but an adult would. Namely a revanchistic coup against Zuko to reinstate either Ozai or Azula and restore the Fire Nation empire, while Azula was redeemable in theory Ozai wasn't, my source for that last bit being that he is voiced by Mark Hamill.

Point is that leaving him alive while idealistic and honestly pretty gnarly also leaves loose ends.