That misses the point. Gender roles is a cause for anxiety because the cost of failure is magnified and strike at the gender identity of the person.

Without understanding gender roles the different in anxiety in fathers and mothers is unexplained, but there is no reason to do so as gender roles is a fairly well documented and understood concept. Even in households that do not perceive the father as the provider and the mother as the caregiver the gender roles in society still imposes expectations that results in anxiety and stress that align identically to those gender roles.

I am not sure where you got the concept of a competition, as the key insight from identify the problems of gender roles is that both women and men suffer from it. A simple decision such as taking a day off to nurse a sick infant should have same outcome for both women and men, but with gender roles pushing the mother to prioritize the child over work and the father to prioritize work over the child the outcome becomes stereotypical.