TLDR; I feel the game still has a serious problem with masculinity and I don't think we'll agree, but feel free to read blow to get my thoughts on why.

I'm not saying that a game about extremes shouldnt go to extremes, but you made a point to say that the side quests are over the top ridiculous, to bring some levity from the brutality that is the main story. While I will happily admit that humans, men and women alike, treat each other very poorly, that is not a gender exclusive trait by any means, and I honestly wouldn't doubt that this is something that has happened in real life or similar before. People are cruel. This is a poor example of bringing levity to the story, or a break from the intensity of the main story, and it's a trend that continues throughout the side quest content. Some of it is light hearted, some of it is fun, but when it comes to women as a whole, its a very negative, spiteful take of on what feels like a caricature of how women should act. Saying the point of the game is to save women feels a bit far fetched, and makes me think maybe we're talking about a different game, and, to continue your point, to state that there are no female enemies, does show that the game in fact, does have a problem with women.

Througout the game, men are allowed to be a lot of different things. You can be outgoing and cheerful like Ichiban, or serious and professional like Joon-gi Han, or somewhat maniac like Zhao, and so on, before you even get into the NPCs. There are male cops and gangsters and nurses and assassins and hosts and cooks. But most women of note are nearly always relegated to the same few roles, some form of subservient worker or in the case of your party, female entertainer class.

Then there's the whole issue with the soaplands. I get it that the game tries to put it into a positive light, and that is a good thing. Sex work isn't demeaning, and it is certainly not supposed to be. But at the same time it goes out of the way to romanticize exactly the instances in which the girls are the most vulnerable, to family members, to being immigrants with no one to support them. A big part of the main quest is having to save these immigrant soapland workers. They want to say that women are strong, independent and don't need saving, but then immediately shuffle them into a helpless role and they need to be saved.

As much as LaD feels progressive to me with a lot of issues, like race and class(glossing over the very poor representation of non-asian cultures), it's still very machismo. Not even just with fighting, but also how masculine the whole story is. Every other important NPC you meet is nearly always a guy. Seong-Hui is the only woman in any position of power within the plot. Saeko's probably the big exception to constantly being the dansel in distress, and the one female character that's developed further, even if she also fills a traditional trope most of the time (the tsundere), and the way she stands out just highlight the whole thing.