Male and female minds and bodies are optimized for somewhat different purposes. Our distant male ancestors tended to hunt and fight more, while females tended more to gather and care for kids. For example:

Researchers tracked men and women from a rural village in Mexico as they foraged for mushrooms. … Men were less efficient–they traveled farther, went higher, and exerted more effort than women for the same amount of mushrooms. Women also collected a greater variety of mushrooms from more sites. This pattern is consistent with the theory that, during the hunter-gatherer period of human evolution, women honed spatial skills needed for gathering while men honed spatial skills needed for hunting. (more)

Now sports let us show off many kinds of physically-expressed abilities. But it seems to me that most sports emphasize hunting skills, such as chasing, evading, throwing, and hitting, far more than gathering skills, such as visual search and fine finger control. Now it makes sense for men to prefer hunting sports, but oddly females also seem to prefer them; pretty much all sports emphasize hunting more than gathering skills. Why don’t women prefer sports designed to show off the skills for which female bodies were designed?

Now men do seem more keen to show off than women, who seem more keen to observe and evaluate. So we should expect to see more men than women doing sports. And if the fixed costs of creating a sport were high enough, there’d be only one or two sports to play, and they might all be tuned for men. But this hardly describes our world.

Men also seem in general to have more skill variance than women. So if only a small fraction of people, the very best few, played sports, we might expect most of them to be men, even in sports that emphasized gathering skills.  We might prefer sports that show off male skills best, if would be mostly men playing no matter what the sport was. But in fact most people, including most women, play sports, at least during their school years.

So why do both men and women prefer sports that emphasize male hunting type physical skills, over female gathering type skills? Looking for parallels, I notice that women are said to look good in male-style clothes (e.g., suits), far more than men are said to look good in female-style clothes (e.g., dresses). Women also earn more respect succeeding at male-dominated professions than men earn by succeeding at female-dominated professions.

The general pattern in all three cases is that we seem to respect women doing well at what mostly men do far more than we respect men doing well at what mostly women do. For better or worse, male abilities seem to more define which abilities count most for high status. Doesn’t seem fair to women, but there it is.

Added 12a: Yes there are activities that are like gathering.  But to be a sport, an activity must be scored and publicly ranked.

Added 5p: The main puzzle is school girl sports, as adult women do far less sport. The main alternative would be to make fem kids be physically active, but in some more fem like gathering way.  Perhaps this is part of how schools acclimate kids to being ranked – the quick and easy way to do that for girls was to make girls compete in male sports.   Inventing competitive gathering type sports would have taken a lot more work.