In fact, many trad heads I know don't actually regard the Ancient Music of Ireland as being part of The Tradition at all, which is something I've never really fully understood.
If a Scotsman resident in Northern Ireland has problems understanding this, I'm sure a lot of people from "across the water" do, too!
Perhaps the best way to understand this attitude is to realise that "the tradition" is a misuse of the language. "The" always implies that there's only one of what we're talking about: THE Pope, THE United Kingdom, THE North Pole. So "THE Tradition" implies that there is only one tradition. Even if we qualify it as "THE Irish Tradition", it's still misleading, because Ireland is big enough to harbour several traditions, even in the narrow area of music.
Not every Irishman is a Roman Catholic Co. Clare peasant. Protestant areas have different traditions; Cork, Antrim and Dublin have different traditions; middle and upper-class Irishmen have different traditions; and urban traditions are different from rural ones.
Jigs and reels were not so important for city people, who could go to dance halls with professional bands. Unaccompanied ballads were not so essential for people who had pianos in their drawing-rooms. This is not to say that the middle class was any less patriotic, or "irish", than the rural labourers. The Irish drawing-room of the early 20th century was filled with the strains of Thomas Moore's very Irish ballads, and Herbert Hughes' arrangements of Irish Country Songs for voice and piano. These remained popular in Ireland long after similar-style songs from across the water had been forgotten.
And the urban proletariat had their street ballads, and comic songs borrowed from the music halls. All this was folk music, and even the originally composed songs took on the nature of traditional songs, being sung by people who had never seen either their words or their music written down.
All these different traditions were traditionally rather narrow. My parents - born in the 1st decade of the 20th century - are a case in point. My mother was born in Derry City, and her violin lessons were aimed at getting her into the symphony orchestra. My father was born in rural Co. Derry, and his fiddle lessons were aimed at getting him to play for village dances. My mother always regarded "comeallyes" as the very epitome of primitive barbarism, whereas my father would often sing this kind of song when he was alone. Both were Christians, and enjoyed singing sacred songs - but again, my mother would sing the rather more classical Wesley hymns, whereas my father would sing what he called "Gospel comic songs" - the kind of popular, evangelical ditty that came over from America.
I am told that my parents performed a duet once - once and never again! (Nowadays, they wouldn't have been welcomed at the same sessions, I suppose!)
It's all a matter of perspective, really. From the outside, Ireland looks small, compact and homogeneous. From the inside, it's criss-crossed by a network of fine lines that are, however, distinct enough to make an Irishman on the other side of them seem different. Europeans and Americans - and to some extent the British, too - who have latched on to a certain genre of music which is definitely Irish may be excused for jumping to the conclusion that this is the be-all and end-all of traditional Irish music.
ITM, for instance, is a tradition. But only one of many. If it has developed so as to exclude songs and instrumental airs - that's fine. There are enough other traditions that include songs and airs.
If we can use the definite article with "THE Irish tradition", then it is the tradition that different regional and social groups each have their own notion of "Irish Music."
BTW, I wouldn't include Carolan in traditional music either. Traditional music is handed down; ancient music is written down. AFAIK, Bunting's written notes are the only access we have to Carolan's works. The Clare peasant of two generations ago didn't know them, nor did the middle-class Dubliner. I do, however, play Carolan pieces in recitals of Irish music, because Irish they definitely are. My preferred instrument for them is the 5-string finger-style banjo - sounds closer to the harp than the concertina does!
Hope this helps,