Taking into account my limited understanding of the language, read on:
In this case, the kanji being used for "kata" means "one of a pair", or that "it" (the set) is incomplete. In Japanese, nouns alone can have both a single or plural connotation depending on context, meaning "tsubasa" can mean "wings" just as much as it could mean "wing" (though I'm guessing the former comes to mind more naturally for speakers of Japanese). Further clarification is necessary, hence katatsubasa, meaning "one wing".
Ichi can't be used for any number of reasons. One is that "tsubasa" conjures up the image of a pair of wings, rather than a single wing, I guess. Think of "kata" as halving that pair.
I may be wrong, but I hope I helped a little.
Edit: I just realized this is in Intellectual Discussion. Haha.
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