The traditional sports were literally invented, there was a "rational recreation" movement during the runup of industrialization when the ruling class were looking for ways to build teamwork skills among workers and discourage sedentary games with drinking and gambling.

Do you think the sport of baseball just emerged naturally from the Earth? No! It was intentionally designed from the ground up to increase productivity and work ethic, to teach workers to follow a coaches orders, to enforce a rules-based mindset, and to encourage competition. Patriarchs invented baseball.

We can make new sports, like your example paintball, that don't have this patriarchal baggage and that everyone can play without needing hierarchical or biological separation. We don't have to play the same sports forever.