I can't find a reference to a Crusades connection. The phrase seems undocumented before Shakespeare, Romeo and Juliet (ca. 1592), which is commonly cited as the first reference in print. It appears that the phrase refers to a sixteenth-century form of horse racing where a lead horse was released before the competitors who then needed to chase the moving target, after the fashion of geese in flight, following the lead goose. See also this reference.

It's not uncommon for people to forget the original meaning of these sorts of phrases and then to provide their own meaning. The folktale "The Golden Goose," ATU Tale type 571 (also known as "All stick together") features a fool who comes to possess a goose made of gold. People try to steal it or its feathers and then are stuck to it, one to another, forcing them to walk together, following the fool and his goose wherever they go. Perhaps you are linking those two motifs - the story and the phrase "wild goose chase" - or perhaps others have made that link (but I am not able to find that link outside of your question).

Hope that helps.