Nope again! Proto-agricultural enclaves such as Catal Huyuk existed for thousands of years with no leaders and a completely egalitarian social system. It eventually eroded, but mobility is not the key to egalitarianism, nor are very small numers such as the H-G bands that only had 20-50 members. Catal Huyuk was a community of up to 10,000.
"This culture was not a lone outlier in an otherwise warlike and stratified world. Rather, it was a snapshot of what the world had been like until that time, when increasing population density as well as greater personal property that came with a larger reliance on agriculture combined with natural disasters and incursions from more warlike Proto-Indo-European tribes changed the social dynamics forever."
I've been researching this and writing about it for over two years now. I've essentially done a self-directed Master's degree on it, and it's one of my areas of expertise. You can keep floating theories, but I have the research and the stories already written to keep shooting them down.
What you are passing along is the propoganda of patriarchy trying to justify itself. I don't blame you for believing it. It's what we've all mostly been told, but it's not accurate or true by any demonstrable metric.