A guy that I worked with in the 80s was a retired Navy SH-3 helicopter pilot.  Early in his career in the late 50’s, he had to transition from Airship to helicopter.  He flew the K-class Airship

And yes, one of the requirements to be qualified as a K-Class Airship pilot was to be able to take a pigeon, put a message on it, and throw the pigeon out the window in the correct direction and with enough velocity so that it did not go through the propellers.

He did not mention the wrap, but that makes sense as well – delaying the pigeon from spreading its wings until after it was past the propellers.