It sounds to me like the cutthroat system OP describes would cause the rise of the sort of people who would perpetuate systemic misogyny.
If by "the real problem" they meant "the root cause" then their theory does appear to fit the facts. I personally suspect it was "both" rather than "one caused the other" but have insufficient data to go beyond "suspect".
The jacket drama had nothing to do with gender really and she was making a big deal out of something trivial. I can imagine the scenario is like "ok we're too cheap to create custom jackets for everyone so we will order the jacket size that fits the majority and the minority can try to fit in it". If the gender ratio was the other way around, there would be a similar situation.
The response to other reports are standard. Whether it is a homosexual boss or a female boss, the company will seek to protect itself.
Sure, and we could save money by only have women's restrooms on every other floor, or not at all since we can certainly save money by not hiring women at all. Its just economic sense: we know that its easier to find men, so lets just optimize for men. Then there's all that diversity training. And don't forget women create tension at the office because they are pretty and their women parts distract the men who can't be expected to control themselves.
If you think that any of this is trivial, you are part of the problem.
Your argument isn't relevant for the situation and is very black and white. Wearing male sized team gear is a trivial matter. There are obviously other cases where accounting for every minority is important but this is not one of them.
I do get that it's inconvenient but it's a huge slap in the face to tell a sizeable percentage of your workforce that no, you can't get them a jacket that almost everyone else now has. They could've at least given them something of equivalent value. Even if it might not be sexist, it's a dick move.
If by "the real problem" they meant "the root cause" then their theory does appear to fit the facts. I personally suspect it was "both" rather than "one caused the other" but have insufficient data to go beyond "suspect".