There is a lot of fat in the cuts of meat used for birria so that may give it the texture that some people think of as greasy. If you want less fat, you could make it ahead, remove (or shred) the meat and store the sauce in the fridge. The fat will solidify on the top, which you can easily skim off. The broth may not be quite as rich this way if you are planning on using it to fry the tortillas in.