Agreed. Just like the example of Cynisca of Sparta shows, when she won the Olympics in chariot racing in 396 BC. The Olympic games were male dominant as most of the disciplines favored physical strength. But in chariot racing, where skill was more important, she could win over her competitors by the virtue of being better than them.
About the diversification of labor. It was probably what worked best. The ultimate goal for every tribe was survival I think. So what served this best was implemented or the tribe died out.
For example hunting. While this was a very dangerous job, I see no reason why women woulnd´t be able to do it. But. Because it is a very dangerous job, it has a high mortality rate compared to other jobs. Men have a low value in respect of procreation. Women have a high one. You need a few men but a lot of women for a tribe to survive if we only take into account population replenishment. So if women would do it in high numbers, the impact of their loss on the numbers replenishing ability of the tribe would be grave compared to the loss of the same amount of men. So it would be reasonable to presume, that those tribe who shifted the more dangerous jobs to the more expendable sex, had a higher chance of survival on the long run.
But there are a bunch of jobs which need to be done in day by day. So if a man goes to hunt someone needs to gather. Hence, men went hunting and women gathered. At least that´s how I think the diversification started.