Why have issues of "women's true natures" been so consistently connected to discussions of women's sports?
I think this issue can be seen with the somewhat cruel and unfair discussion surrounding the Williams sisters, whose physique has been mocked and derided for being too manly. I can't remember which one, but one of them played a match recently in a kind of negligée, deliberately designed to show her as "feminine."
This is because athletic endeavors have typically be seen as the complete opposite of what women are "supposed" to be like (by their "true natures").
For about the last 180 years or so, there has been a widespread belief that women are not suited for public life and competition. They are supposed to be nurturers who are best fitted to stay home and out of the public arena. They are supposed to be emotional rather than physical. Women's participation in sports goes against all of these ideas.
When women participate in sports, they are competing. They are acting in a "masculine" way by trying to dominate others instead of being compassionate and nurturing. This calls into question the extent to which a woman who is playing sports is "really" a woman.