The South Korean government doesn't seem to be doing much about misogyny. Until they get serious about that, a lot of young women are just going to opt out. The same applies to many other developed countries, to a lesser extent.
Misogyny, like any form of large scale hate and persecution is strictly bad, and it's strictly good when society decreases it. Not being Korean myself I'm not well-positioned to comment on the current status quo or what the government could be doing about it there.
In America we've made enormous progress on institutionalized misogyny even if it's a bit of an embarrassment that it's taken so long. Women didn't even have the franchise at a federal level until 1920, losing to the Emirate of Afghanistan by a year. And we've had some reversals on reproductive rights recently, it's not a monotone improvement.
But broadly by most any metric you care to choose from college graduation rates to life expectancy to likelihood to be incarcerated to likelihood to commit suicide: women are doing better than men on a lot of really important things.
I don't mean to imply that there isn't a capital-P Patriarchy: there is.
The meme that is destroying this conversation is in considering any man a patriarch, when almost by definition a vanishing minority of men are. An old white guy who is the CEO of a big company or holds public office? Probably almost certainly a patriarch, probably almost certainly enjoys absurd privilege at the expense of others.
But a typical white guy in 2024? He can't even take care of himself in a system that has no use for him. He's in no position to oppress anyone.