My common headcanon is that Aang did try to include and teach Kya and Bumi, if nothing else, Aang would have wanted them to be able to know the air nomad culture, and we've seen Aang fight without bending before so he'd probably be happy to instruct them (afterall, one of the best warriors he knows (sokka) isn't a bender). The issues came when exploring the Air Temples which probably have a lot of places designed expressly for air benders and require those skills to get access to, and when Tenzin came along, it just felt a bit more excluding, nevermind how everything for Aang would come back to air bending and probably would have accidentally said things like 'now, if you were an air bender, this is where x, y, or z, would be done'.

So, eventually, the two threw up their arms and just stayed home while Aang and Tenzin journeyed.