I don't think it's a wilful misunderstanding. I think for Natalie it's an inability to do what she wanted JK Rowling to do. She wanted JK Rowling to listen and take her fears seriously. But she didn't offer to listen and take JK Rowling's fears seriously in return. She didn't offer to put aside, even for a moment, her foundational belief that life is so much harder for transwomen than it is for anyone else, for long enough to hear how bad life can be for women, and for long enough to hear how getting what Natalie wants could make life harder for those women who are just as vulnerable if not more so.
Because if Natalie had to accept that so many women are vulnerable in ways that transwomen aren't and that women's fears are genuine and founded on fact, and these fears are not simply the result of failing to listen to Natalie and failing to realise how much more persecuted and deprived transwomen are than anyone else, then Natalie might also have to accept that she really can't have everything that she wants, or at least not ethically.
But if Natalie was able to listen without bursting into tears of self-pity then maybe there would be room for negotation. And then maybe it would turn out there's a way for Natalie to get the things transwomen most need, the healthcare and the protections, without needing the automatic access to everything that protects women. Protection as a transwoman. Well, that's what I hope anyway.