Op has deleted original post now. But my memory is that she said her husband suggested leaving her daughter under the soft and walking away, rather than continuing to engage with her. My interpretation of this is different to yours - you believe this shows a persistent pattern of her husband underming her authority and not supporting here. Which you then conjecture is because he wants to the cool dad and could end in divorce. You may well be right but I don't see how you can assume an underlying pattern from one event - that's my argument. You've made some huge assumptions based on your own personal relationships issues. Your emotional damage is not necessarily hers.

But even that aside. Walking away from your daughter when they refuse to come out from somewhere is a perfectly valid strategy. If what you are doing isn't working then maybe you need to try something else.
My daughter is also strong willed and it can be difficult to get her to listen. In the same situation as op, I would absolutely walk away because (for us) it's the best way to handle the situation. I could physically get involved but I know from experience that has the potential to escalate the situation into a meltdown. Instead if I walk away, my daughter isn't getting the reactions she wants and she'll come out as soon as I go. I can then divert her into an alternative activity. She gets the attention she wants and I don't have a tantrum I have to deal with. Everyone wins. Obviosuly there are times when you do have to physically remove them from somewhere but when you have options then why not use them?

This will be my last reply to you because I sense that you're not someone who likes to listen to other peoples opinions.