I disagree with your view, and here's why: There are many cultures that were not based on the "Adam and Eve" story, but instead had other creation stories for their culture. Some of these stories involve women in a more elevated way. However, matriarchies (women-led cultures) have been very few and far between.
The subjugation of women (and children, and the handicapped, etc.) has always been due to perceived vulnerability and weakness. So, because in ancient times there was a greater need for physicality in day-to-day living, men were perceived as superior due to their greater physiological strength. Also, in the absence of any enforced social order, MIGHT actually does make RIGHT. Without enforced laws, for example, stronger people can mistreat weaker people and benefit from it. I don't think it's shocking for anyone to say that the history of humankind is the history of the pursuit of self. If you are only concerned about meeting your own needs and desires, you won't hesitate to do anything to make that happen--including manipulating and subjecting the weak. This, btw, STILL happens. Laws cannot force people to be moral. For example, while it's *morally right* to pay workers a living wage, insure safe conditions, etc., there are plenty of companies that engage in legal but immoral shenanigans to increase the profit margins of a few at the expense of the masses. You even (unfortunately) see this within Christianity as well. The vast majority of televangelists make their exorbitant incomes off of the backs of the poor, sick, and disenfranchised.
However, when you look at the bible as a whole, it is very clear that God has a heart for the weak, poor, sick, and disenfranchised. His agenda is for these groups to be cared for by those who are stronger, richer, and more able. Part of the measure of any civilization (if you look through Christian lenses) is how that culture takes care of its most vulnerable members. This is why there are verses about how rape should be handled, for example. (Verses that might seem harsh to modern readers, but that were actually exercising justice at that time). There are lots of examples of God elevating and valuing females in the bible--ultimately culminating in the birth of the Savior from an impregnated woman. Jesus consistently valued women in his ministry. The NT has lots of examples of women actively working with the Apostles to build the church.
In my opinion, the reason we have laws protecting the women and the weaker members of society is a manifestation of the Christianizing of the world. (The WHEAT does grow with the tares). For example, children were once considered the "property" of parents. Parents could mistreat and abuse children in many ways without consequence. Now, we have lots of laws to help protect children from abuse. The same is true for women. Women used to be property owned first by their fathers and then by their husbands. Now, women (in the Western world, especially) have political agency, they can own property and businesses, they have access to education, and can achieve comparable incomes to males in most fields.
If a person's philosophy is that women are less valuable than men, it's easy to misquote or misinterpret scripture to prove that. However, that really has nothing to do with God's clearly revealed perspective. So, if someone wants to stand on "Eve sinned first" as a reason to malign or mistreat women, ultimately that person will stand before God to answer for it.