I suspect patriarchy was instituted because for most of history physical ability was incredibly important. Societies were based largely on farming, which required a great deal of physical strength. This likely put men in a position of power.

Also, being physically stronger than women meant that women really had no ability to argue with the situation. It's only with modern technology, guns for example, that women are able to effectively defend themselves against an aggressive man.

Well, I don't know all the details.

Interesting theory, but I don't think it's correct. I highly doubt that equality has ever really been tried at any other point in human history.

Thanks for reading :)