>...it has always been about allowing those who are socially superior to avoid the embarrassment of having to compete in an environment where they might be defeated by their social inferiors.

Is your argument actually that women don't generally compete with men in sports because the sports don't want to embarrass the male athletes if they lose? If so, I suspect this is a bad faith argument, but if not, you can simply do a little searching to find that there is often quite a bit of difference between the performance of top tier male athletes and top tier female athletes. For example, no woman has ever run a 4 minute mile in competition and more than 2,000 men have and even about 30 high school boys have. I am sure you can find other examples.