In defense of "The Unicorn Song," let me say that I think it's often accepted as Irish not only because an Irish band made it a hit, but because it SOUNDS IRISH.A reflection of Shel Silverstein's gift was his versatility. A Los Angeles Jewish guy writes "Queen Of the Silver Dollar" (PURE COUNTRY), "Beans Taste Fine" (with good BLUES flavor), "Unicorn" (very IRISH motif), and quite a few children's poems and songs, as well as serious articles for adults. (I doubt that I know the half of his repertoire.)
The point is that composers/songwriters who become familiar with a genre can often write very convincingly in it, without the benefit of having been born to it.
I realize that lots of Irish bands are sick of playing it (having run it into the ground, as it were), but that's a different matter.
Genie