There is a very pronounced prejudice in discussions dealing with patriarchy. It is viewed as an evil institution to be destroyed. On the other side there is a big women are wonderful effect when discussion matriarchy.

Both are simply two sides of the same coin. A woman's restrooms is neither morally better or worse than a man's restroom. Patriarchy just means a society dominated by men. Reality is, many women, even when given equal rights to vote and hold office, want to vote for a man. (For example voting for a male president when there is a female candidate). There can be many reasons, but we know this is based on self interest. Maybe the male candidate is seen as more capable, has good leadership, instills a sense of security, etc.

Another example is if a woman decides to marry a man who is dominant, a leader, has a higher status (economically, socially, politically) than her. Then by default in every aspect of life naturally she takes a follower role, unless the couple discusses things to be equal. It seems to me that women generally do not complain when a good capable man who cares for her takes a leading role in the relationship. In fact its a problem when he doesnt take enough initiative. Of course there are exceptions, but we are speaking in general bell curve terms. Loosely relevant figure, but studies show that 75% of men are dominant while 4% of women are dominant within the BDSM community.

There are many cases of women submitting to patriarchy voluntarily and there is nothing wrong with this when women are free agents who make choices based on their self interest. Men can also submit to matriarchy voluntarily but the fact is, there are not as many women willing to do for a submissive man, everything a dominant man would do for a woman. (For example be the breadwinner while the man stays at home. Again, exceptions exist)

One might ask, why Patriarchy when we can have Equality? Women say they want equality but this is the biggest instance of "actions speak louder than words". Insisting on gender equality as a man who is dating a woman, is the surest way to breakup. Men and women are complements, not competitors. I say, let people decide for themselves what they want from life, and dont judge them for it or seek to fix something that isnt broken. Seeing a low proportion of female CEOs, presidents, etc is not a gender issue if seeing so many more male homeless than female homless is not a gender issue. Gender wage gap is not a gender issue when even female managers prefering to hire men is not a gender issue.

In conclusion social progress should seek to eliminate clearly abusive and toxic practices that infringe upon people's freedoms and well being. For example, domestic abuse, sexual assault, etc. Voluntary agreements entered upon by consenting adults in a free market liberal democracy should not be the targets of social justice. We should be asking whether we are seeking equality of opportunity or equality of outcome.