So the segregation of private and public spaces is now equivalent, is it?

Er.. no. There are spaces that are privately segregated (that exist unchallenged because of some very clearly limited legal exceptions) and there are publicly shared spaces. Confusing one with the other in order to enforce private religious sensibilities upon others in public spaces is disingenuous at best, as the issue is not about stopping people from self-segregating.

Go through the guidelines and substitute the word “race” in place of “gender” and use “black” and “white” instead of “men” and “women” and see what you get. And if segregation on racial difference is unacceptable so too is it on gender or is that your aim as well? By all means self-segregate but do it in private, in the public arena we are all equal, irrespective of someone’s “genuinely” held religious beliefs, whatever that means.