The definition of "woman" has always been more fluid than "person with two X chromosomes". I agree that it's not *arbitrary*, but it is to some extent socially constructed. Reasonable people have always allowed for edge cases that don't fit the category neatly, whether someone born intersex or someone who just chooses to sincerely adopt the opposite gender role.
It seems to me that the current controversy is caused by some people insisting that we don't have the right to decide for ourselves who is performing which gender role while other people insist that gender and biological sex are the same thing—both of which are errors, IMHO.
What's unfortunate is that the extreme positions of each group tend to create more of their opposite in reaction. I suspect, for example, that J. K. Rowling's position would be much more nuanced if it weren't for the unhinged reactions to her, IMHO, reasonable concern about women's safety being compromised by a minority of bad actors exploiting the insistence that anyone who doesn't accept all claims at face value is a bigot. Likewise, trans activists aren't wrong to think that some of their opposites are motivated by bigotry.
I guess I'd just like to see the sane majority insisting on nuance and refusing to "take a side" in this unhinged culture war.