Part of the issue is that, in patriarchal societies, women are viewed as the "weaker sex". In past ages, they were viewed as less intelligent, inferior, and were not permitted to participate in political or religious debates. Those stereotypes still exist despite science proving otherwise. It took decades of effort just for American women to gain the right to vote in 1920.
I see the women's sports issue as an extension of that. No one is crying because trans men are in the locker rooms, showers, bathrooms, or competing and winning competitions against cis men. Why not? Because, using their own logic, their egos cannot handle the idea that they got "beat by a girl." But they also think of women as being weaker and needing protection from predators who would take advantage of them. So the sports/bathrooms are the perfect place to focus the crosshairs.
It doesn't matter (as we have read above) that studies have shown that the advantages disappear while on hormone therapy over time. Even within our own community, there are people who believe the stereotype, or even embrace it as a desirable trait. It doesn't matter that statistics show that transgender women are not misbehaving in restrooms, are not committing mass shootings and other violent crimes, and, in fact, are more likely to be the victim in a confrontation or criminal act.
Media, especially social media, is a political propaganda machine. It is used every day to spread false and misleading information to shape public perception for an agenda. Politicians do it, advertisers do it, organizations do it, and the public eats it up and helps to spread it around.
So what is the source? If we step back and take a hard look at the core of what "genetic advantage" implies, it is inherently racist. The Far Right™ is clearly a White Christian National movement. But due to the backlash from prior civil rights movements, no one wants to be labeled as a racist. And yet, civil rights were denied to a major chunk of our population based solely upon genetic and cultural differences.
The "genetic advantage" in sports is a false flag. No one is suggesting banning East Africans from running marathons, despite the fact that they consistently win due to cultural advantages and a willingness to train extensively to enhance them. No one is suggesting banning other races from sports due to their biological advantage, despite medical testing, historical studies, and other evidence that they have a "genetic advantage".
When we allow governing bodies to use that criterion to strip civil rights, well, as the saying goes: "Give them an inch, and they will take a mile."
That's my take on it.