OP, it's the generation. My mother has anxiety issues and can't do much on her own. She'll never sit in a coffee house or bookstore alone. It makes her insecure. The most she'll do is go to Bloomingdales by herself. Our generation tends to handle being on our own better than the previous ones. Did MIL go to college and live on her own before marriage? Neither of my parents were on their own, EVER. Went from their parents' house to their house owned as a married couple with nothing else in between. Not even a college dorm. Nowadays, plenty of women have a lot of practice living alone and aren't afraid of it. We buy homes on our own, we travel on our own, we drive country on our own.
IMO, you should probably try to see her as a product of her time and treat her like someone from another country who has very particular customs they must adhere to. No sense in arguing with her. As far her anxiety and how to impacts you and your family, tell your husband if he wants her to stay with you, he has to handle her issues. Other posters are right: she is NOT going to go to therapy. She thinks her fears are justified and anyone who disagrees is being the unreasonable one.