To add to squids answer: There isn't a segregated mens and womens category. There is an open category and a women's only category.
What happened in the open category is that because the societal pressures and social constructs that disincentivized young girls to play, women weren't placing high in the open category. (Because top players end up being top players because they started when thwy were 5) This leads to a feedback loop where young girls see less women in the sport and get reinforced that it is not for them so don't pick it up at a young age, so less persue it and get good, so less women are seen at high levels, etc.
So then comes the women's category to combat women not feeling like they belong in that space. Women can compete in both the open and women's categories.
But because it is an intellectual thing mostly, barring transgender women is ridiculous. In athletic sports you could almost try to argue that a woman that went through male puberty could be stronger(ignoring how estrogen weakens them and they cannot compete in the men's category anymore). You could try to make that argument in athletic sports (and it is a different discussion to this) and almost seem logically consistently on the surface level if you don't think about it any further than your fox news talking points, but what is the argument here? If a woman went through a male puberty they were possibly socialized as male and weren't told as a kid that chess wasn't for them and so they have an intellectual advantage over cis women?
I don't get it. It seems like, just with athletic sports, it is not about the sanctity of the sport or about fairness, it is about banning trans people from public spaces and policing what women can be.
You mentioned the US, so please look up the FCC rules regarding limiting tenants from putting up antennas and dishes. It might be illegal for them to restrict that.