Do just what you suggested. A new back and binding will keep it from stretching and tearing and make it useable. You can take a picture of what it looks like now, in case anyone is interested in it's history. Also make a new label and indicate it was rebacked on that. I think I am more hurt when some I gift with a quilt doesn't use it, than if they use it and it gets worn. (I want my bed size quilts to be providers of warmth and love, not cold museum pieces.)