Not really Unbridled. No movement for change has ever had a majority actively working for or identifying themselves with it. People have always fought to maintain the status quo, people have always blamed the oppressed for their position within society.
As someone who is American, I find the thing about the draft argument a red herring before we get into this story being British story so bringing the US into is grasping at straws. No one in my generation has ever been drafted, those who have been don't blame women for it, the selective service system was not put in place nor is its continued presence maintained by women. This system was put in place by rich men to bring in poor men for slaughter, they do that better now by promising health care and education. If men organized to get the requirement removed, it likely would, but then what would they have to beat feminists and young women with? Women fought for many things, including sex separated facilities because we were being attacked regularly in public, including facilities to help domestic violence victims and refugees and those who are suicidal. My local women's centre which is fighting for funding and only open two days a week due to this is still helping a local men maintain their low-cost counseling services because without them there would be none at all.
Right wing women are not an entirely different species from left. I come from the Bible Belt, I know women who voted for Trump who hate my existence and would tell me I deserve to burn in Hell forever if I wasn't already shunned by them, I know women in the UK who voted UKIP (I live in one of few areas that still has UKIP councillors) and love to tell my immigrant self how we're ruining everything and they hope I get kicked out and I still have more in common with them than I do many of the left wing men and women who also regularly throw those they dislike under the bus. I certainly have more in common with a right wing woman who fights for single sex facilities than the left wing ones who like to tell my feminine son that he can "grow up to be a woman" and tell him he's wrong to get upset about not being seen as the boy he is, I have more in common with the UKIP right wing parent who agrees with me that boys who sexually harass girls should be restricted from the park than the left wing one who wanted to blame everyone but the significantly older boys who made porn-inspired sexual remarks threatening girls as young as 7, some of whom were attacked. Girls who were attacked because of what those boys thought of girls and their bodies, just as women are (and I don't know where you are Ruth, but everyone I've been, women have been attacked and treated like freaks, attackers who feel we should be responsible for their views on our sex). It's not how we identify, it's what we do and who we protect and who we fight for.
Just as women and disabled people fought and continue to fight for facilities, dysphoric and sex role nonconforming people have been fighting for sex neutral preferably single stall facilities for decades. I remember it from many moons ago when I was an awkward guy-passing dysphoric teen. You can hear . The very recent push by a very loud few to just let everyone go in whatever sexed facilities won't help the most vulnerable, they will continue having difficulties and attacks while those who already can muscle in will cause harm and push to take up more and more space from those who can't risk fighting back. The people pushing for self-ID and such really don't need us to defend them.