Here's the thing about the transgender movement, which goes beyond simply fairness in sport, although that's a part of it. There's a central contradiction at the heart of the whole movement's philosophy (if you can even call it that), which simultaneously renders it totally unworkable, but also makes its adherents even more zealous in spouting it.
On the one hand, it takes its cues from "classical" leftist sources--feminism, the civil rights movement, gay rights, etc. It's largely a legal argument, not a philosophical one. It imagines transgender people as a minority of the population that has been oppressed and ignored for far too long, and now demands their right to participate fully in public life "as their true selves" (to use a phrase from a certain very successful middle distance runner who subscribes to this). This argument necessarily depends on the idea that trans people are not the whole population, not even a large part of it. They are a class of people, a class which is demanding their civil rights.
But on the other hand, when you try to press the point and ask questions like "what is a woman?" or "who counts as trans?," the rhetoric shifts from the legal/civil to the philosophical/ontological (or destruction thereof). Gender, they say, is on a spectrum. There is no male/female binary, only our "performance" of them. These are the Judith Butlers. But here's the thing--if gender is a spectrum, then everyone is potentially a trans person. It doesn't matter if you're the paradigmatic example of your own gender, like a Grace Kelly or a Paul Newman. You're just performing your gender, even if you don't realize it. In this formulation, trans people cannot count as a separate class in any meaningful sense, since everybody is included--it's just those who choose to embrace the spectrum. It's all arbitrary. Furthermore, if it's a spectrum, why is there any significance in transitioning? And if we decide gender by a person's "innate inner sense" as opposed to his or her genitalia, isn't that exactly as arbitrary as deciding it based on genitalia over innate inner sense? It's all arbitrary.
The point is, it is either 1 or 2, but they want both. It can't be a set legal class/category AND ALSO an arbitrary Heraclitan river of flux.
These two sides are irreconcilable, which is why I suppose many love to embrace and evangelize it.