It's a pretty firmly established theory that patriarchy and plowed agriculture arose at the same time. Here's another quote that speaks to why from a 2018 article in the World Economic Forum addressing how the rise of plowed agriculture also led to other broad cultural shifts:

“Labor roles became more gendered as well. Generally, men did the majority of the fieldwork while women were relegated to child-rearing and household work. Without contributing food as they once had (and by association, without control over it), women became second-class citizens. Women also had babies more frequently, on average once every two years rather than once every four in hunter-gatherer societies.”

“Because somebody had to have control over surplus food, it became necessary to divide society into roles that supported this hierarchy. The roles of an administrator, a servant, a priest, and a soldier were invented. The soldier was especially important because agriculture was so unsustainable compared to hunting and gathering. The fickleness of agriculture ironically encouraged more migration into neighboring lands in search of more resources and warfare with neighboring groups. Capturing slaves was also important since farming was hard work, and more people were working in these new roles.”

"This division of labor and social inequality had very real consequences. For instance, while the majority of people had disastrous health compared to their hunter-gatherer ancestors, the skeletons of Mycenean royalty had better teeth and were three inches taller than their subjects. Chilean mummies from A.D. 1000 had a fourfold lower rate of bone lesions caused by disease than commoners."

https://www.weforum.org/agenda/2018/10/how-the-agricultural-revolution-made-us-inequal

As I said in this OP, when women contribute to food production in significant ways, they have more status in other things. Places with "dry agriculture" that needs to be irrigated and plowed are cultures with a history of significant gender disparity.