Unfortunately facts don’t matter, because this is all about prejudice. That’s everything you need to know about right-wing rhetorical strategies: Reality doesn’t matter, because the purpose is to stoke up prejudices against “the enemy”.

If enough people hear about trans people taking over sports, something will stick, even if it is nothing but a vague association between trans people and sports. Even people who are generally supportive of trans people will thus become susceptible to argue about trans people in sports. That will reinforce the association and thus further the prejudice.