Is the husband considered at fault for backing down from confronting his wife's family after discovering that her mother is the source of rumors about his family's lack of affection for his wife, despite his wife's continued accusations and pressure regarding his family?

You did well by not answering your wife's demand for divorce. Divorce should not be resorted to except after all means of reconciliation have failed. It is not permissible for a wife to seek divorce without a valid reason. You should be wise with your wife and strive to mend the relationship between her and your family. Explain to her that among the virtues of a wife's character is showing kindness to her husband's family and overlooking their mistakes.