I have to disagree here. Both dd-mm-yyyy and yyyy-mm-dd are much much better than the mm-dd-yyyy. Both are very intuitive compared to the completely unintuitive mm-dd format.
I mean, it's no more arbitrary than any other order. In the context of "why does American English use this ordering", it has the desirable characteristic that it matches how we express dates in our everyday language. A trade-off exists between that characteristic and the desire to have the date go from smallest to largest or vice versa, but both approaches have practical benefits.