I'm here using transphobia to generally indicate a set of policies and outlook that, in essence if not stated outright, seek to remove trans people from public life. The largest part of this is the argument that trans people are inherently a predatory threat and should therefore not be allowed in spaces around women and children. There's a fairly small but coherent ideological group here, with a small number of politicians across different parties (most notably Rosie Duffield in Labour, Joanna Cherry in the SNP, Shahrar Ali in the Greens and Baroness Ludford in the Lib Dems), journalists, and public figures who are tub-thumpingly in support. An organisation called the LGB Alliance is their largest organisational body (and I think all the abovementioned politicians have e.g. spoken at their conferences etc.)
The LGB Alliance's positions include that "protecting children from a dangerous and confusing gender-identity ideology" is important such that any content related to trans people shouldn't be taught in schools, and indeed some of their people take the view that this should apply to LGBT content generally, one of their co-founders citing the risk of "predatory gay teachers" in having school organisations or clubs around LGBT issues. They're generally opposed to the concept of Gillick competency (that is, the idea that teenagers should have some privacy and independent assessments of their capacity to make medical decisions, including from their parents if necessary). They're also against banning LGBT conversion therapy (including for LGB people!) on the grounds that they think the ban on conversion therapy will make it harder to persuade children that they aren't actually trans. This somewhat differs from the view of experts and groups like the UN, who have called for
a global ban. Essentially a lot of their positions return us to the pre-2000s "don't talk about LGBT issues/panic when they're discussed around minors" view of the world, but somewhat more focused on trans people.
They also tend to take a very specific view on what laws discussing sex/gender/etc mean, and their view is that they
should always mean binarily understood chromosomal sex and nothing else, and that the definition of single-sex spaces
should be heavily enforced including in spaces where such designations are generally self-made: so in other words, no trans people in public toilets and changing/fitting rooms. This is explicitly not the view of British case law and equalities guidance (which tends to not take a rigidly chromosomal definition of sex on the grounds that this would be somewhat ridiculous: the formal guidance on implementing the Equalities Act has generally suggested the position that single-sex spaces excluding trans people would be legitimate if in service of a reasonable and proportionate aim: to the best of my knowledge there is very little in law that has confirmed any situations that actually meet that test). Generally the LGB alliance has been veering rightwards in its associations over time, because they get more of a hearing from conservatives, and hard-right groups like the Heritage Foundation in the US will invite their speakers over and be personally nice to them.
So, re your edit, the transphobe lobby in UK politics is far, far closer to the first bit than the second. The sort of hyper any-slip-up-is-aggression viewpoint, from a British perspective, seems like a rare feature of corners of social media and North American campuses, it's simply not something I've ever seen in UK electoral-political circles (I'm sure the UK has a few of those people, but they don't tend to muddy their hands with actual political parties). A lot of the LGB Alliance type people are perhaps politer and better turned out than Jordan Peterson and the most wingnut online fringe (the online wingnut fringe do exist, I've had to block probably several hundred nameless Twitter accounts claiming my pro-trans-rights views make me an abuser, over the years). Nonetheless this is very much a concerted movement on the borders of middle-to-upper-class British society (polling generally suggests that the electorate more widely do not care about trans issues). This group is a minority within the centre-left parties, but it's a pretty well funded minority who really like taking crowdfunded legal action even despite some stunningly sizeable failures - one flagship discrimination case that they touted as a "success" for getting a 22,000 payout on one of its points cost over five hundred thousand in legal fees. One suspects there are some lawyers somewhere who've worked out that this is a pretty reliable way of getting money out of people without having to go to the trouble of coming up with sensible legal arguments. Regarding the GPEW/SGs split, the primary issue is the apparent inaction of the GPEW on the issue: so it's not so much that the GPEW espouses the above ideology, but they've not treated e.g. Shahrar Ali's views and associations as a deal-breaker in the way they probably would do if one of their senior members was espousing other bigotries.
Regarding Pent's post - I think Stonewall tend to make that point about some very young children having trans identities to push back against the "oh people can just choose to not be trans" view which still exists. Care for young people with dysphoria/potential trans identities is a tricky area to get perfect, but I don't see anything wrong with the idea that people should be told that it's OK if they take time to work out who they are in an identity sense, which AIUI is really what the idea of affirming education is about: it's affirming the possibility and taking a case-by-case approach for what works best for how a child is interacted with as opposed to actively working to reinforce a child's birth-assigned gender, not doing the "oh you say you're a girl, small human, YOU SHALL NOW WEAR PINK FOREVER" which seems to be what Shahrar Ali is imagining. I do get that it's easy to fear that affirming care can become reinforcing an identity that's still developing or ends up wrong for the child, but the cases of that seem AFAICT
extremely rare compared to the much more common reverse problem of non-affirming, trying to force people to retain their assigned-at-birth gender, and ending up with a bunch of suicidal trans teenagers. I certainly think/agree that comparing being trans to being schizophrenic is
miles wide of the mark: and yes, I've found lots of people in his kinds of circles expressing the view that being trans is a mental illness and the appropriate care is to convince people that they're not trans (which is another root of the scepticism among this movement about banning conversion therapy). I also 100% agree that Twitter is a terrible and clunky platform for talking about what are obviously really really complex issues that should ideally be treated quite individually and where it's extremely difficult for most people to work out what's going on.
But yeah, I think age-appropriate education that helps children understand what different identities are and how that might apply to them seems very reasonable, and I'm much more worried about the risk of UK going back to a pseudo-Section-28 system where LGBT issues are just swept under the rug in schools. I don't think there should be as much reliance on the charitable sector to be dealing with both advocacy and service delivery in some of these areas, which has maybe been an awkward point for groups like Stonewall and Mermaids where they're both having to be the public face arguing the case for better care/funding and being the groups that do a lot of work to deliver resources for equalities training, helplines for young people, service referrals, etc etc... but that's part of the UK's general chronic underfunding of public services, and I'd fault the government rather than the charities for setting up that situational problem.
Sorry that ended up so long, I'm still feeling like crap and I tend to overexplain when I'm struggling to express myself well. I frankly wish I didn't end up knowing this much about an issue that I would like to be able to not personally care about because I'd like to be able to leave it up to trans people and relevant experts to sort out... I find it really exhausting how much time and effort ends up being spent on this, and miserable how it's ended up as one of these daft 'culture war' things. One of the biggest recent social media storms over this stuff was one of the prominent LGBA types yelling at a local library in the UK and people sending them hate mail over the fact that the library used they/them pronouns for their new reading mascot, who is a colourful dungarees wearing alien. Which is just the icing on the cake of how incredibly silly this has all gotten - but it's a silliness that certain people are spending a huge amount of money and legal time on in ways that then force people like me (activists in political parties) to waste endless time on it. I can't talk about all the exhausting machinations I'm vaguely aware of that underpin some of these things in public because it's hard to remember if there are bits I don't officially know re ongoing legal stuff, but I'll very much admit at this stage that my view of this movement has been coloured for the worse by many of their antics: I've tended to find them entitled in their mentality, obtusely obsessive in their ideology, and aggressively timewasting in their methods.
Meanwhile the UK is still working out what to make of events, and Liz Truss still looks incredibly weak. Journalists have been asking if she is 'a prisoner' of the new Chancellor, and one of her own MPs recently described a speech by her as "the first time I have heard a corpse deliver its own eulogy". And tons of budget measures have been U-turned on. So that's all going well...