No it doesn't. I agree that it's very popular to say so. You can be forgiven, to some extent, for saying so. But you keep putting blinders on to avoid examining things! So I don't see how, given that you won't look at other countries or throughout history (despite using historically-referencing terms like "slightly less"), you could have thoughtfully examined things yourself. But let's have a bit of a look around anyway.
Do you think murder is a problem? According to the Man Box study, the U.S. and Australia have nearly identical levels of support for the hegemonic masculinity end of the act-like-a-man axis, and yet Australia has 8x lower murder rates. And Japan? Patriarchy up even higher, and 4x lower murder rates. Culture trumps "patriarchy" so hard that it's not even worth worrying about patriarchy when it comes to murder, for the most part.
(Aside: gang violence specifically does seem to be driven strongly by hegemonic dominance hierarchy issues, as do some wars--e.g. in Sudan. This is a serious subproblem that needs to be addressed...except it already has been addressed to a large extent between about 250 and 50 years ago, where rule of law and consent of the governed and human rights became internationally accepted in contrast to violence-based hegemonic dominance. Yes, there are still problems, but in places where this has not taken sway you see vastly worse outcomes. But your attitude--that we live in a strongly patriarchal culture--suggests that you see the well-documented widely-implemented solution as part of the problem!)
Do you think public verbal harassment of women is a problem? Japan, which has the worst rate of gender equality among technologically advanced countries, has practically none. (It does have a lot of sexual harassment, but in private.) Culture trumps patriarchy when it comes to public behavior.
Do you think the pay gap is a problem? It shrunk a lot and the (strong) shift towards an adversarial approach that you promote hasn't budget it the tiniest bit; if anything it's getting worse (https://www.pewresearch.org/social-trends/2023/03/01/the-enduring-grip-of-the-gender-pay-gap/). Young women are very nearly at parity, which, along with other studies, indicates that about half of the problem has to do with the differential demands of parenting. (That's orthogonal to dominance hierarchies, incidentally. Other kind of patriarchy.) Capitalism is on at least equal footing with patriarchy when it comes to paying people, but patriarchy is legitimately an issue here.
(Amusing aside: in cases where an unfairness distorts the market, one can employ capitalism's ruthless seeking of market-optimal solutions to drive fairness. That is: if your competitors are under-paying women, just offer women more, and you'll get the best women working for you, and ta-da! You win. On the other hand, if women aren't motivated by money as much as men are, on average, capitalism won't fix the problem...but...we also have to ask if there even is a problem. If some people like money more and some people like other things more, and people get what they want more but those preferences aren't perfectly distributed by gender, is this really a problem?)
Do you think it's a problem that people don't respect women in leadership? That isn't even true. Between Angela Merkel, Nancy Pelosi, RBG, Condoleeza Rice, Madeiline Albright, Margaret Thatcher, there are and have been for decades women whose own accomplishments and personality allow them to command instant respect from practically everyone. Yes, there are a small number of idiots trying to make an issue of it. Practically everyone thinks the idiots are idiots. There is a small bias remaining, which is enough to sway elections tuned to be very sensitive to small biases, and there's a moderate difference in how people express extreme displeasure with men and women. Regardless, positive perception of female leaders has been rising until, roughly, the widespread onset of the more confrontational anti-nuanced position that you exemplify (https://www.forbes.com/sites/kimelsesser/2022/12/15/young-adults-are-the-most-biased-against-female-leaders-data-shows/?sh=18fb96336fe2), most strongly reflected in younger age groups. This is indicative of bad contemporary messaging rather than old reactionaries (who are most easily engaged by backlash), unlike, say, trans rights where young people are much more openminded than old people.
Do you think rape and sexual violence is a problem? As bad as it is, culture seems to provide a profoundly strong protective effect given that people who are outside the normal cultural bounds--people who are bisexual, for example--end up with shockingly higher rates of victimization (https://sexualassaultsupport.ca/statistics-sexual-violence-in-canada/). Whichever kind of culture there is in Canada, it's largely protective. Throwing the label "patriarchy" at something that varies by a factor of thirty in ways that are orthogonal to patriarchy (but not culture) is absurd. Again, culture trumps patriarchy. The situation is absolutely not okay, and "Man Box" attitudes probably are exacerbating the problem, but that isn't synonymous with patriarchy (or with broader culture).
(A lot of (some flavors of) patriarchy is about keeping young men in line not because it's good for the one at the top but because socially unattached young men have a propensity to engage in profoundly anti-social behavior.)
Do you think that domestic violence is a problem? There's a clear role for "Man Box" attitudes to exacerbate the problem. But that isn't just traditional patriarchy because that discourages multiple sexual partners, and multiple partners is a risk factor for intimate partner violence (https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3951472/). So it's a "Man Box" attitude problem that is fought by traditional patriarchy. This isn't to say that there aren't even better ways to fight it, but pinning it on patriarchy is entirely too naive.
And so on and so forth.
Different aspects of patriarchy are responsible to some extent for some of society's problems. For most problems it's not the main thing.