I think you accidently double postedHe wasn't dead that long. I don't think it's an instant reincarnation sort of thing.
if he died, he would have been deadTo the Avatar Community: When Aang died at the end of Book 2, shouldn't have the next Avatar been born? Or do you think it died at birth when Katara brought him back to life?
This. The cycle would have ended, but I believe if he wasn't in the avatar state then another would have been bornRegardless of whether it's an instant reincarnation or not, there wouldn't have been another one because Roku told Aang that if you're struck down while in the Avatar state, the cycle ends. And Azula killed him while he was in said state.