I mean, the easiest way to understand the 'blind spot' thinking of 'but WHY can't they see the bleeding obvious?' is to put yourself on the side of the debate in a different moral/contested opinion, one where you fall on the traditionally liberal side.
Imagine someone insisting to you that all the facts support that, for example, abortion is always wrong, children are harmed by growing up with same sex parents, that there is a god/heaven/hell, or that men are superior to women, there should be no restrictions on gun ownership, or climate change is fake, or whatever, and they don't understand how you can possibly not agree with them when you're otherwise such a sensible person. Would it change your view?
But beyond that, I can offer my perspective as an ex be-kinder:
- ignorance. Not that they are ignorant people, but there's so much about this topic that isn't covered by mainstream news. People on both 'sides' vastly underestimate how little the average person knows (and previously I would say 'cares' but that is changing slightly now) about this issue.
A lot of people still couldn't say definitively whether a trans man is a man who is now a woman or vice versa. The vast majority of people I've spoken to assume that all transwomen get their penises removed.
The over focus on toilets by the media means that so many other issues, the ones that (mountain-top)ed me, aren't common knowledge - things like transwomen medically producing "milk" for babies, rape crisis centres, the % of transwomen in prison for violent offenses, pretty much everything on the terf is a slur website, etc. The converse of that is that the 'facts' that are bandied around without qualification about the high percentage of trans murders, "most vulnerable" 'it's this generation's section 28' etc. they do believe, because again they haven't seen them contested, and have seen them repeated by people they trust/admire (celebs etc).
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Not being personally impacted - most women won't ever go to prison or know someone who has, so the possibility of being locked in a cell with someone with a penis, doesn't really matter to them. They aren't competitive athletes so don't really care whether women or men come first in a sport they don't care about. The number of transpeople is so tiny they don't really see their lives being impacted in any negative way, so don't understand why GC people are making such a fuss.
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Personal experience - they have trans friends/family members whom they like/love. They know their trans friends would never want to hurt people, so they assume all trans people are like them, and therefore should be treated kindly. Nobody wants their friends to be discriminated against.
They might genuinely see their trans friends as their 'new' sex. Posters here insist that you can always tell (and I agree often, maybe even usually you can), but I know 2 trans men, one of whom I classed as a really good friend, and I (and the rest of our friends) had absolutely no idea until they told me. Even now I hugely struggle to think of them as anything other than males in my head. I would feel utterly ridiculous calling them 'she' and, again, despite the majority viewpoint on MN being that 'we welcome transmen into women's facilities,' realistically I know if my mum or gran or whoever saw my friends in the women's loos they'd be really freaked out. Because they look like men, muscley, beardy men!
- They honestly don't care. Again, lots of posters on here insist everyone cares about the importance of female only spaces really and are just being handmaideny or performatively woke or whatever but there are women who honestly are completely not bothered about sharing services, changing facilities etc with men, trans or not. Perhaps they have a very queer or just mixed sex friendship circle so are completely used to sharing bedrooms on holidays, etc with male friends. They don't see any difference between a gay male friend, a gay female friend or a trans friend - they'd happily share a bed with any of them.
Lots of young people travelling (me included) actively choose to stay in mixed sex hostel dorms rather than splitting the friendship group. We all genuinely preferred to be with men we knew than women we didn't. It's not exactly the same but in lots of countries in europe that are a lot more relaxed with nudity than us, people legitimately don't bat an eye about being naked in a sauna or topless on a beach with people of the opposite sex.
When GC people insist that women only spaces are needed because women are a) vulnerable and b) want women-only spaces they think, well, I'm a woman and I'm not and I don't, so if they're wrong about that they're probably wrong about everything else too.
so lots of reasons, and of course 'be kind' and 'being on the right side of history' play huge parts too.