I wish I had seen your post before diving into my own research project on these patches, Mike J. I can add what I think is a cool detail to the conversation.
With regard to the brown uniform background actors: the U.S. Army 27th Infantry Division insignia has an interesting design history. The 27th was formed from the New York National Guard (hence the NYD – New York Division – you can pull out of the line work), and during WWI under the command of Major General John F. O’Ryan, who’s role was memorialized with a play on words depicted in the stars shown on the insignia: the seven primary stars the make up the constellation Orion.
I believe that the use of these patches is purely coincidence (based on the evolution of that famous line, and the production timeline), and I have not found any discussion regarding insignia’s design in any of my research to date. The serendipity, though, makes me happy.