Misogyny, resentment and subterranean norms Can you actually make that argument rather than asking us to assume it? — unenlightened
In the Bronze Age, the most important commodity, food, was not private property. Land wasn't. People worked in the fields and brought their produce into the temple to be divided by the priests. It's called a temple economy. There was no free market. How patriarchal were they? We can only speculate. In the opening scene of the epic of Gilgamesh, the people of Ur are praying to the Sun god to help them because Gilgamesh, their king, is making all the men work hard, and he's having sex with all the women. The Sun god hands the problem off to a female divinity, the fertility goddess. She makes a man out of clay and sets him roam like a wildman. In due course, the wildman is tamed by the temple prostitute. This wildman eventually becomes the best friend and homosexual lover of Gilgamesh.
So we have a bisexual king, the one who initiates wildmen into a civilized state is a prostitute. One of the most important deities is female. In real life, the leader of the temple, where the food is divided up, is the King's daughter. I'm not suggesting that women had equal rights in this society. I doubt anybody had any rights per se. But this is not patriarchy as we know it.
The conditions you describe for the genesis of patriarchy, where private ownership drives men to know who their offspring are, didn't exist until the Iron Age. Our knowledge of the Iron Age is not foggy. We know it pretty well, and though a case could be made for what you described, if would be fairly flimsy.
All we know is that as the dust cleared from the Bronze Age collapse, patriarchy had become normal. From early accounts, we know this was a very dangerous world to live in. No one travelled around. You just stayed close to your clan.
So maybe ownership played a part. Maybe patriarchy became the dominant cultural scheme for other reasons. I'm speculating just as you are.