>> With this knowledge, what reason is there to classify these individuals as female?
We already do: we determine the sex of babies at birth, and each others' sex as adults, by looking at external genitalia and at secondary sex characteristics.
Why would knowledge of chromosomes now change that? Like I say above, we can't see chromosomes. And the people who can, i.e. biologists, have a much more nuanced view of what it means for an individual to be male or female, rather than their karyotype (and not their chromosomes) being male or female. The scientists interviewed for the BBC article all agree that whether an individual, a person, is male or female, is not clear cut and that detecting, or not, a Y chromosome is not enough of a criterion.
Keep in mind that "all X have Y" is not equivalent with "all who have Y are X". Or replace "all" with "most" and the same goes.
>> I think that is more to do with socio-cultural views on women and men than anything else, and not so relevant in the narrower case of eligibility in women's sport, which has to be based around eliminating male physical advantage, otherwise there's no reason for the category to exist.
The BBC article points out that there is no test that can determine "male physical advantage" as you put it. Also note that some athletic associations, e.g. World Athletics, consider athletes with Y-chromosome DSDs to be eligible for competition in women's categories as long as they reduce their testosterone to male levels medically. See:
https://www.bbc.com/sport/athletics/65051900
As an aside, note in the same article that World Athletics now completely bans transgendered women _who have gone through male puberty_ from competing in the women's category so even having one's sex determined as male as birth is not recognised as an absolute advantage in women's sports; the cutoff is having gone through a male puberty, which a genetic test alone cannot determine.
To be clear, I do think that banning trans women who have gone through male puberty from sports is _on balance_ the fair choice, although it means transgendered women cannot reasonably compete in any category, since it would be unfair for them to compete in the male category and there is no special third category. End aside.
>> But isn't it also cruel to female athletes when they are being compelled to compete with a male?
Yes, but I'm claiming that someone whose sex was determined to be female at birth, and lived their whole life as a woman, and experienced everything that every other woman experiences, is not a male.
>> My view is that sports organisations have a duty of care and a duty of fairness to all athletes, not just the ones who've received unwanted news about eligibility due to their sex.
Of course, but those athletes also deserve a fair treatment. And ruling that they can't compete as women, when they are recognised as women in every other aspect of their lives since birth, is very hard to see as fair. Should we start to treat those women as men in their everyday lives also? That sounds even more unfair.