Looking through my recipe collection and the internet baking sites, icing sugar isn't that uncommon a choice. The Fanny Farmer Baking Book (1984 ed. by Marion Cunningham--damn but I'm old) has three versions: one made with confectioner's sugar, one with white sugar, and one with brown.

My favorite recipe, which I got from the daughter of a patient I took care of (I still have her handwritten copy), uses regular sugar but adds a quarter cup of cornstarch/4 cups of flour. It tenderizes the cookie a bit, I think, and I assume that's why some bakers (and eaters) prefer using icing sugar.