Don't treat Normal as the "bland" type. In the pokemon universe it has its own special characteristics, powers, and weaknesses. It makes perfect sense for, let's say, a commonly seen lizard that has electric abilities to be Normal/Electric or a fluffy bunny that's trained in the ways of combat, but still retains its "earthly" characteristics, in terms that it resembles real wildlife, to be Normal/Fighting.
Aparently most Normal pokemon do not have a second type and most that do, it's flying due to the wide array of birds that exist without special elemental traits.
If those do not convince you, the metagame needs to be balanced, so Normal deserves much like any other type to have dual-typed pokemon. Logical flaws will exist as the opposites mix, but that does not limit the game from including such pokemon. Primary examples are Malamar (psychic/dark), Gliscor (ground/flying), and all the possible combinations of ice/fire, fairy/dark, water/fire, dragon/fairy etc. If those make sense, then Normal duality has no reason not to exist.
edit: On topic - I would retype Chatot as pure Flying, because it makes sense for it to be more of the manifestation of a musical note, but I can still see it being Normal. Flying is quite an abstract concept that few ideas capture it completely, when trying to create a pure one. Being a cloud (fakemon that never makes the cut for some reason) is enough excuse for it to be pure flying. The same goes for beings that rely on the weather, like Castform, which I believe needs to be retyped as Flying.