Should a teaching statement always be included in faculty job applications, even if not required?
I don't think you need to include a teaching statement if one is not explicitly requested.  However, you should be very clear in your application materials about what courses you want to teach and/or are willing to teach.  That is usually the information we are looking for when we ask for that document.

Including a teaching statement when one is not requested is probably not going to be a negative (most applications we get have just been sent out in bulk, and many places require one) unless someone on the committee is exceptionally nit-picky.  But if that is the case, do you really want that person as a colleague?