I think the author of the tweet is correct in one sense - that sex-based rights do create division, because they assert that women (and men?) are entitled to certain privileges on the basis of sex. That's an anathema to the concept of sexual equality in the West, which asserts that men and women should be regarded equivalent social actors.
GC feminists argue that women and men should be treated differently in certain circumstances, that women have rights that are exclusive to them on the basis of their female biology. I'm not saying they are wrong (I agree with them), but I think we have to acknowledge the tension here with traditional liberal feminist narratives about women and men being 'equal' and therefore basically interchangeable.
The truth is, men's strength, and thus their potential for violence, is much greater than women's. Only one sex can kill the other with their bare hands, so in this important respect the sexes can't ever be equivalent social actors. Because this is a biological fact, I can't see this changing, although we can mitigate it through laws and social norms.
So women do need certain spaces and services to ourselves for reasons of safety and privacy, and obviously we need our own sporting competitions. But modern notions of equality make this difficult for many women to admit out loud, and I'm not sure that 'sex-based rights' is the right meme with which to advance this argument, because so many people think the progressive thing is to ignore the salience of sex altogether.
The phrase itself, 'sex-based rights', is potentially an alienating one for young women who see their claim to equal personhood as being rooted in people not noticing their femaleness. And perhaps not only young women, as this goes all the way back to the Second Wave.
Ruth Bader Ginsburg's feminist legal career was about challenging exclusion and discrimination 'on the basis of sex'. The point of this argument was that sex doesn't matter - the fact that women have different bodies to men should be irrelevant to how we treat them in society. Catharine MacKinnon extends this logic to prisons, arguing that female-only prisons constitute a form of sex discrimination against men (google MacKinnon's interview with the Transadvocate for this - I won't link to it as the site owner collects info on GC women).
In MacKinnon's reasoning, 'sex-based rights' shouldn't exist, not even in prisons, as they are by definition a form of sex discrimination. And they are, of course - but the counter argument would be that because of our bodies, women shouldn't be treated as equivalent to men in all circumstances - there are some instances in which the differences between male and female biology need to be taken into account; thus, certain services and spaces which are exclusive to females.
To return to Ginsburg, she is the person credited with introducing the word 'gender' as a replacement for 'sex'. She did this on the suggestion of a female colleague, who thought that it would make her feminist legal arguments more acceptable. Using 'sex', she reasoned, might put the judges in a lascivious frame of mind, whereas 'gender' is anodyne, with no messy associations with the body:
“I owe it all to my secretary at Columbia Law School, who said, ‘I’m typing all these briefs and articles for you and the word sex, sex, sex is on every page,’ ” Ginsburg said.
“Don’t you know that those nine men (on the Supreme Court)--they hear that word, and their first association is not the way you want them to be thinking? Why don’t you use the word gender? It is a grammatical term and it will ward off distracting associations.’ ”
It's ominous that in order to win legal equality for US women, lawyers like Ginsberg felt it was necessary to linguistically decouple women from any 'distracting associations' with the body. This elision of the significance of the sexed body is now well established in feminist thought and disseminated throughout the mainstream - it makes consciousness raising around 'sex-based rights' an uphill task.