Sexism inflates the perceived agency of men and deflates the perceived agency of women. This means that people take men more seriously than women when it comes to things they do or things they're responsible for. Women, on the other hand, because they're seen as lacking in agency are viewed as vulnerable, so while they're not taken seriously when they act they are taken seriously when they're acted upon.
This, to me, is the fundamental dichotomy of sexism.
When I say that women are given a pass by sexism, I mean that women's decreased perceived agency leads not only to them being taken less seriously when they do something positive, but when they do something negative as well. Take, for instance, the sentencing gap which gives women a significant discount in terms of prison sentences for the same crimes as men. The reaction to women abusing men is also massively different from the reaction to men abusing women. Domestic violence services also tend to discriminate against men. Some abuse hotlines even forward men who call them to batterer's hotlines. Some men are even arrested when the police show up to their houses after they've been abused.
These are symptoms of a society that does not respect female agency or male vulnerability.
So how does patriarchy theory tie into this? By emphasizing sexism instead of attempting to eradicate it. What does patriarchy theory say about society? It suggests that all or nearly all the power in society, as if power were a straightforward and easily quantifiable finite resource, is in the hands of men. Not just individually, either. It's quite demonstrably true that most of the richest people and most of the government of the United States (and probably many other places) are male, there's really not much argument to be had there. What patriarchy theory does, though, that goes beyond that is to suggest that this is demographically significant to men who are not, in any sense of the word, "powerful". It looks at the top, but it fails to look at the bottom. It doesn't realize or doesn't care that nearly all of the people who die at work or at war are men. It doesn't matter that men are overrepresented in prison or in the homeless population (especially the long-term homeless population). None of that, by the way, seems to be harming all those rich male leaders of government and industry. They step over the corpses of the working class regardless of gender, though statistically mostly the males.
That, to me, is the first reason I'd say patriarchy theory is sexist. It's based on the premise that the most downtrodden men somehow benefit directly from the most successful men as if gender created a cohesive division of interest among the species by which all benefit from their "own" gender "winning". Basically, it's a sports rivalry but everybody keeps the mascots in their pants. I'm not into it.
But what you asked was why I said patriarchy theory perpetuates sexism or contributes to it or something. Not just why it is sexist, but why it makes sexism worse. Well first, you can easily see, I'd think, how building your narrative around dismissing half of your problems with sexism out of hand could be construed as not doing a great job of combating it (the first rule of patriarchy club is that gender must be a distinct power stratification with women on bottom) . More importantly, though, patriarchy theory is basically just sexism copied and pasted.
Unless you disagree with all that stuff I said about what sexism is in the beginning there (which maybe you do!) then it's pretty obvious, I'd say, that patriarchy theory is just basically agreeing with sexism. Genders addressed as distinct groups with distinctly different needs, abilities, and outlooks? Check. Men have all the power and not much vulnerability? Check. Women are super vulnerable and need protection from a world in which they have almost no agency? Check.
That might be great if you live in a country where the laws are actually built around oppressing women. As in, you're not allowed to vote or you can't drive a car or you can't own property or people stone you to death for getting raped. These are things that most certainly do exist in the real world, and in those contexts I'd say patriarchy theory at least makes a lot more sense. In America, though, where I mostly hear about this idea from, women are overrepresented on college campuses. Most of what's left of the wage gap is made up of dangerous jobs that women don't want and insanely demanding schedules that women (in what I personally think is the intelligent decision) tend to reject. There are still a few states that have some really backward laws about birth control and abortion, but that's about the only element of what you might call institutional sexism that's still codified in law in the United States.
How, at this point in the game, does it benefit anyone to pound away at this narrative that women are being held back by sexism in a way that men just aren't? Wouldn't it be better to eliminate all the unnecessary contention from anti-sexist thought and activism by not kowtowing to a narrative that's never really gotten the whole picture and that in many places is so antiquated as to be more of a hindrance than a solution?
To me the solution would be to encourage people to treat one another the same regardless of petty demographic information. I mean, that's the goal, right? We want everyone to be treated based on their character rather than mere biological chance. That's what I want, anyway.
TL;DR (I hope not): Treating men and women as if they had levels of agency and vulnerability as per sexism's assumptions is itself sexist. We should be trying to combat it by treating men and women as if they were the same in terms of agency and vulnerability.