It is a very interesting debate, and I don't see any reason that there isn't a pure Flying type, other than the programmers just didn't make one. I think that pokemon such as pidgey are normal/flying because most of them are based off of real-life birds, such as pidgey being based off of a pigeon(hence the name, pidgey...), which are fairly common or 'normal'. I think probably the main reason that there are no pure flying types is because most all of the other pokemon that are part flying are based off of something else that calls for another type (ie. Gyrados is a flying sea serpant thing, flying dragons, ect.), and they haven't really had many that couldn't be classified as some other sub-type.
As for Fighting type, I think it is not referring to a state or even will to fight, but rather a particular style of fighting. Most fighting pokemon and moves are generally associated with some kind of martial arts, and include kicks, punches, and other moves that are used in such practices. Hence why a lot of trainers that have fighting-types are karate kids and such.