Not sure how I missed this thread...this is my jam.
I've done birria a few times with venison. Along the way, I did a little bit of research on the dish and there seems to be some thought that birria was originally made with vension. My guess is that it was a way to cook the desert mule deer that can be a little tough/lean. It is also traditionally made with goat (same reasons).
The key for really good birria was to add some bones (usually beef or pork) bones to the broth while the meat is cooking as they give the consomme some body. The broth/consomme is what separates birria from barbacoa. Toasting all the chiles and roasting the onions further adds flavor and complexity.
I am just as happy to eat it as a stew as I am making tacos and dipping them in the consomme.