Well, no, that's not how nature works. The natural world works on overlapping niches within which there is some competition, but it's not that chaotic like a free for all.
What makes a hominid "fit" is adaptability; otherwise, we'd still be someone else's dinner. We're not the strongest. We're not the fastest. Arguments could be made about the cleverest, but that's only within environments we control. Most people, in a deserted island situation, would not be able to make it without rather immediate rescue. So how clever are we really?
Perhaps there's some hubris in that biggest, strongest, fastest mantra?
Besides, the strongest is rarely the fastest. The fastest is rarely the most resilient. The most resilient is rarely the cleverest. The cleverest is not the most persistent or diligent. And so forth and so on.
Besides that, what difference does it make to species B that species A is stronger when species B eats bugs and species A eats grasses? Why compete over something you have no idea for? There's less competition than you'd think, given the focus on that by mostly male naturalists in the late 19th century that has continued forward. Competition is not king.
Further, men and women are not competing species. We are the same species. Some of you gents reading this might need to take a minute to go get your smelling salts. Your argument is flawed here because you've drawn a false comparison.
It is unnatural for social species to compete against each other in all things. There is some competition during breeding season for mates. But you're typically not competing for food.
Besides that, as omnivores, hominids have a very very wide range of food sources. Our ancestors ate a whole lot more than aurochs.
And no, I don't think Patriarchy is natural or inevitable. I actually think that in a lot of eays it has stymoed our evolutionary development because it eliminated more natural forms of competition and replaced them with what we have now, where competition leads to one winner rather than the betterment of many.
Patriarchy came about because of an environmental upheaval. Before that, we more stable as hunter gathers for hundreds of thousands of years. What Patriarchy has done is put is unending competition for resources and wars. It's also created a population explosion that cannot be maintined indefinitely. We've been in a cycle of boom. We're about to enter a cycle of bust.
Exponential growth will eventually lead to exponential decline. Ending patriarchy would have helped the drawdown not be as hard or (dare I say). cataclysmic. But I think it's too late for that now.