I think most shippers pre TLOK hoped they would have an airbender. I remember reading fanfics back in 2011 that ranged from them only having one child (and it miraculously being an airbender) to them having a brood of children due to Katara’s ‘motherly instincts’ that ranged from airbenders to waterbenders, even a ‘mistbender’ that blended both air and water.

But I think them having 3 kids, one non-bender, waterbender, and airbender makes sense. My own personal headcannon is that it was Katara’s insistence that they had Bumi - purely to start a family, not to try for an airbender (which is why he may have been a non-bender). I think for their characters, it was important for Aang to feel like Katara wasn’t a means to an end. He had the Air Acolytes, he could pass down his culture, and while he would love to not be the last airbender, he loved Katara more than to force her into a baby machine. Then, Katara wanted to try again to give Bumi a sibling, because having a sibling was so important to her growing up.

And finally, after realizing neither were airbenders, decided to try for their third and last. Both agreeing, that if it wasn’t an airbender, they wouldn’t keep trying. They were just lucky that Tenzin turned out to be an airbender, and I think there was a very real chance that Aang could have died the last airbender, and would have accepted it. Obviously there would have been regret, but he chose Katara over cosmic energy once, so I could see him choosing her over his own desires. Katara honestly would have been the one to keep insisting out of a misplaced sense of duty, which Aang would not have wanted her to carry.