I appreciate you voicing your opinion on the issue.
The first item which I would note is that I do not need to prove a negative - you need to prove the affirmative.
When discussing the patriarchy, most women would describe the patriarchy as a system that delivers an inherent advantage to men. I note that this system (if it exists) should not be producing outcomes that are so disadvantageous to men. There is additional evidence in various pieces of psychological and sociological literature that contradicts the affirmative that there is indeed a patriarchy. Another example is the difference in outcomes produced by the court systems (I have cited the evidence in my article "No, I will not use your Pronouns"). Women have a systemic advantage in criminal and family courts.
Indeed, it seems that this system continuously produces outcomes that are not beneficial to men and often benefit women. To reiterate, this does not mean I believe in a matriarchy, but instead that there will always be differences in outcomes between men and women. Some outcomes will benefit the former, while others benefit the latter. In a patriarchy where men control the system, it does not make sense why they would be enforcing a system that is harmful to themselves.