It’s tough ain’t it. But… you have to develop an understanding of what your wife is feeling and work with her, not at her. You have your theories about what will solve the problem. They may or may not be correct. But you won’t get anywhere just telling her how to do things. Especially if she isn’t that type of character.
From your original post – Asking her to calm down and picture what will happen to the kids in years to come if she doesn’t stamp down some authority is going to pile on the pressure to her, make her feel like the worlds worst mother and create tension between the two of you.
Everyone has there own parenting styles. The two of you need to sit down together and work out how you are going to get through this.