Well I do know that one, why sex but not race and it's because of confusion between sex and gender. People can alter their gender-based ideas and performance, unlike their race but also not their sex. Though interestingly, while skin colour is biological, 'race' is very much a social construct (a deeply embedded, culturally laden one that cannot be casually messed with, respectfully), so more similar to gender than perhaps we'd previously recognised. The only inherent and immutable physical phenomenon in that group is sex.
I think a lot of this issue and the natural, lightly held 'why not be kind and considerate towards the ordinary, harmless, 'real' transwomen' attitude, that so many people, who haven't thought the subject through, hold, is based in nice people assuming that other people are as nice as they are.
This happens all the time, in all aspects of life. Nice, well-behaved people attribute good motivations to others. Bad motivations and unthinking bad behaviour, just don't occur to them. Whereas selfish, manipulative and aggressive people assume that everyone else is as scheming, unpleasant and oppressive as they are and use this assumption to justify their own actions.
You see it on here, in all topics of conversation, all the time.
It's a really important thought to hold in mind, in any situation; 'This person's motivation and normal pattern of behaviour might be quite different from mine. Therefore I cannot predict the way they will behave next. If I want to understand them, I need to step back, listen carefully to what they're saying, look at what they do, then draw inferences from that gradually, perhaps considering well-known behavioural patterns and statistics, without overlaying it all with a thick layer of my own wishful thinking.'
The other general point, usually relevant to the relationships board, that I think is relevant here is 'never feel sorry for someone who has power over you'. Well, you can feel it but don't let it influence your decision-making.
I think that's the basis of the trick here - performed knowingly by the TRAs, innocently by ordinary trans people - perpetuation of the idea that trans people are universally the most oppressed people, more so than women, always and in every situation. So the flipping of normal ideas about who has power over whom.
Yet these ultra oppressed people retain male privilege in a patriarchal society. They are also the subjects of pity by other men, people who do hold more power, so can legitimately feel sorry for them and act upon that.
These seem to me to be the reasons why their wishes are accommodated in such a knee-jerk way. While other, far more disadvantaged groups, like people with disabilities, have had to campaign for decades to get (still inadequate) access to suitable loos, transport, buildings and jobs. The oppressed have always been oppressed, whereas transwomen are perceived by men, and perceive themselves, as falling from a great height, privilege-wise, so deserving to be caught, by everybody beneath them.
If you think about it in terms of power, you realise that in some situations, women are always below male-bodied people, never above them. Therefore, we have to protect women's sex-based interests and keep our empathy for trans people abstract and available for other, harmless, situations.
Of course the biggest, maddest TRA trick is portrayal of the view that people cannot change sex and that sex is sometimes more important than gender, as one that belongs to 'feminists'. When it is in fact the normal, everyday view of most people; of the man in the pub, on the bus, in the football team. They're just not interested in talking to those people, because they do not hold anything the TRAs want (except the power to grant their wishes). Far more productive to pull the wool over their eyes, then plead oppression.
Thus, JKR is portrayed as having said something terribly transphobic just for writing down what most men and women think. That lie has been pushed so hard, so deliberately (just like the harmlesses of smoking, the unreality of climate change were in the recent past), that most people, without the time to read everything, believe it unquestioningly.
Then... when people do read what she said and recognise it as normal and uncontroversial, they arrive at the very uncomfortable feeling that their view is not welcome in wider society (rather than in fact, only among the 'pro-smoking, pro-fossil fuels analogous' TRA lobby).
That's why Jo's been such a heroine. She's chosen to speak out, to point out that in fact, the emperor has no clothes.