Okay I'm sure plenty of people figured this out so I might look a little silly saying this but I first watched ATLA when I was a child and didn't understand much other than "bending elements pretty."
After watching it again as an adult and analyzing some plot points I realized that the fire nation killed all of the Air Nomads, then moved on to the Water Tribe and specifically killed every water bender they could, then were beginning to move onto the Earth Kingdom by imprisoning earth benders, and also wanted the Avatar dead (not captured, DEAD)....... because they wanted an Avatar on their side.
They were trying to bring an Avatar reincarnation into the Fire Nation as soon as possible by wiping out every bender they possibly could, and they started at the Air Temples because that was where the next Avatar was guaranteed to be.
They wanted a Fire Nation Avatar that they could train and indoctrinate to fight for them and help them take over the world. They wanted their own living weapon, and realizing that as an adult after not understanding why they were killing everyone for so many years was very eye-opening.
I think the writers of ATLA did a very good job at plot building and story telling through hidden messages rather than having characters outright say everything they were planning, because at no point does the Fire Nation (that I caught or remember) outright say they want the Avatar in the Fire Nation. I originally thought they just wanted him captured so he couldn't get in the way of whatever they wanted to do. But after analyzing and thinking it over, it's very clear that's what they were trying to do; kill every Avatar reincarnation until one was finally born into the Fire Nation.
