She has a point, although I seem to remember most of the candidates were women in Labour's 2020 leadership election.
It is notable that the party which cares much more about identity politics has only ever managed to elect white men as leaders. Perhaps by focusing so much on the sex (and race) of candidates they trigger unconscious sexist (and racist) biases in party members, including stereotype threat among female MPs.
I have noticed in recent years that the left, and especially the unions, are chock full of a particular type of misogynist men. It's quite different from the misogyny on the right - these bros will pay lots of lip service to feminism (or their version of it) while being utterly vile to any woman who does not toe the line. Whether that's 'structural' or simply a large concentration of misogynist lefty bros, I don't know.
I have mixed feelings about AWS. I'm not generally in favour of quotas but when Blair used them to facilitate the election of a huge number of female MPs I think that did reach a sort of critical mass that changed the culture of parliament and the perception of the electorate.
I don't think there should be a quota for Burnham's cabinet. I think Labour just need to sort out their misogynist lefty bro culture and then focus on merit.