Essentially every tech firm is run by a man, yet, they were not awarded those jobs because they are men. They tended to get them by founding those companies successfully, or being very early employees who did major successful initiatives. The US President has always been a man, but that's because the men always win the votes, not because the competition is tipped in their favor.
On the other hand, in some countries there are now laws forcing women into corporate boards. You have to have women there even if they did nothing to deserve it.
> It’s institutional misogyny, it’s morally bankrupt, and it needs to change
This is so far from reality you need to do some grounding exercises. Institutional misogyny would look like written down, formal and openly acknowledged mechanisms that systematically block women from certain opportunities. Can you write down a list of such things? Probably not because in the west, no such mechanisms exist. The same is not true in reverse, where there are actually many such mechanisms that specifically grant women those opportunities by banning men from competing. Board seats, all-women selection lists in politics, recruiters told to only search for women, jobs that are earmarked only for women, laws that grant women child custody by default, the US VP being selected because it had to be a black woman and so on.
People like Andrew Tate get audiences because it's completely accurate to say our society is institutionally misandrist. To claim the opposite is to simply replace reality with ideology, which is always dangerous.