Nice Tony Benn! (Even I know some of his quotes.)
You're totally correct in that if Americans really want change, we technically have the tools to do so. However, money ultimately has a lot to do with winning elections here, and there is big $$$ behind this agenda. Plus, most Americans still feel very tolerant about extending special rights (including to protected spaces) to trans individuals. They have done an excellent job of portraying themselves as truly victimized, oppressed people. (You don't hear much talk about the staggering number of women killed by domestic violence, but being denied access to a bathroom will make the national news.)
You know, I think because the law hasn't yet been fully defined, (federally), it's hard to say what would happen to private "female" sporting clubs. Some cite the Equal Protection clause of the Constitution as a means to support the idea that discrimination, even for private businesses or organizations, is illegal. And I suspect if one lived in a very liberal state, there would be pushback and local challenge in the courts. Living in Utah or Idaho? Might be able to get away with your private group.
I predict that within several years, there will be a case that goes to the Supreme Court and will establish a more definitive basis.