I have some thoughts on this but I'm not sure I can really articulate them.
So - I'm cis (queer, not-white, female blah blah - we do exist on here). I'm not trans. But by chance most of my adult life my main friendship group has been disproportionately trans and gender-queer - so I get some of their viewpoints. People are starting to get the idea that a single voice is not representative of some kind of whole group. Trans people aren't a big coherent group. Viewpoints vary a lot. I'm cis-woman and my voice is in no way representative of cis-women. Here are some things I think about:
My friends mostly fall into white, middle-class, geek-tastic professionals. They mostly have financially comfortable lives with supportive family, friends and co-workers, or at least mostly haven't been disowned. However, if there is a broad observation, it's that being trans is to live in fear for your life. However lovely your friends are, you never know when you're going to meet someone who thinks you should not exist - that you are an aberration, and that fear of erasure is always close at hand. Maybe not every minute, if you're lucky maybe not every day, but probably. That fear is of physical as well as psychological violence. Not all of them have experienced that violence first-hand, but they know - we all know - what the stats are like for murder and violence if you are trans. My friends live with this fear, despite being the 'privileged' ones - not in poverty, not in hostile communities, with decent relationships/family situations.
That idea - of a close-at-hand existential fear - has been recently spread and understood by people who never 'got' it before about the everyday anxiety of being black and just existing. It's also in my understanding of Jewish communities - who some point at as exaggerating dangers - having a cultural shared fear of total erasure, informed heavily by collective trauma.
Trans people and women in general share this everyday fear of violence, although the motives are slightly different. I would suggest that a lower proportion of cis women than trans people carry this close-at-hand fear on a day to day basis but that it's still significant - I have no facts to back this up. I know I don't live in fear. But I also hear about it frequently enough that I trust it's a lot. We also know the stats on murder and violence of women in general.
Most trans people don't want to have these conversations. They just don't want to feel even less safe than they do now. They don't want to feel like the media is encouraging people who might want them dead, who think they're sub-human.
Most trans people wish 'society' didn't think of them as trans at all. They don't want us to go on about it. They don't want us to treat them differently as the person they are. There's a bizarre idea in the media that you can just transition on a whim. It's really really hard, takes a huge amount of guts and bravery, and generally, several years (in some cases decades) of being pretty miserable before you get there.
The sticky thing boils down to:
(Most) trans people have a real and justified fear of violence from society in general (but mostly men).
(Lots of) women have a real and justified fear of violence from mostly men.
These people overlap: trans-women have a real and justified fear of violence.
But also - some women have a real fear of violence from trans-women. How many? It doesn't seem to be common, but I would not dismiss it. Is the fear justifiable? Fear is real, even if not justifiable statistically - but what if these fears are being stoked by the media and by people who say trans-women are actually deviant men?
There are a lot of people speaking up on behalf of other people in an angry and dismissive way.
The toilets thing - I'm pretty sure that women really don't care. Public toilets are not the sacred 'safe spaces' the media make them out to be. I also think we're approaching the problem the wrong way. The problem is not the people - it's the gender-segregated shared toilets. Increasingly new builds and refurbs are swerving the problem completely by just having toilets. For anyone. You don't have to share. It's no-one's business what your gender is because you aren't sharing a space with them. The more this becomes the norm, the better. (I do realise all public toilets aren't going to transform overnight but it could help the way we think about it.)
Most women-only other things are a distraction (sports etc).
The difficult one is actual safe spaces - refuges and shelters. There are two reasons often given: that some women who need refuge (mostly, from men) will not feel safe in a space where there are people around who have not lived as women all their lives; and that some women have a pathological fear of people with 'male' traits which might remain in trans-women. I have no truck with the argument that trans-women are an actual predatory danger in these spaces, but I do think that fears are important. So what is the answer for this?
Firstly, I think this should be recognised for what it is - a specific and difficult area where everybody who seeks space in a refuge is vulnerable and in need of care and protection. To me that means that general laws about gender recognition and equalities should not be argued on this situation but on the wider day-to-day society - which mostly should not and does not care.
And then - is there a way of approaching it with it not being the people that are the problem? If all you talk about are 'women's shelters' like they're all the same thing, and the only thing available, then there is going to be an unsolvable problem - either you have women who can't access refuge when they need it because they fear the trans-women within, or you have women who can't access refuge when they need it because they have been excluded on grounds of being trans. Neither is acceptable. But there are already specialised shelters - for Muslim women, for example. Women with children. Women who don't want to be around children. Why can't this be solved the same way? Generally women's shelters for all women. Specialised shelters for cis-women only. Specialised shelters for trans-women only. More budgets for shelters is what we need. Oh, and less shittery and violence from society (mostly men).
this isn't really about Starmer.