After Margaret Thatcher’s death in 2013, important Labour voices explained absolutely no value attached to her allegedly symbolic achievements as a female leader.

Glenda Jackson implicitly rebuked Ed Miliband, who had noted of Thatcher that “at each stage of her life, she broke the mould”, becoming “the only woman in the cabinet when she was appointed in 1970; and, of course, the first woman prime minister”.

So what, Jackson argued. Since Thatcher was wrong about everything, her triumph did nothing for women, not even in principle. “To pay tribute to the first prime minister denoted by the female gender, OK; but a woman? Not on my terms.” On Twitter, amid extravagant crowing about the witch being dead, the Labour MP Ian Lavery confirmed that female trailblazing is not, axiomatically, an advance. “It would be cowardly,” he announced, hours after Thatcher’s death, “for me not to comment on today’s news. No tears from me nor the mining communities destroyed by MT we R #theenemywithin.”

It has since fallen to Harriet Harman, whose own career was stalled by in-party misogyny, to establish Thatcher’s eternal irrelevance as a role model. She told the Guardian’s Simon Hattenstone how, as a young MP, with an infant in her arms, she spotted Thatcher, whom she considers “an aberration”, approaching. “She looked as if she was going to come and admire the baby. I had this terrible feeling of thinking, ‘I don’t want her to look at the baby,’ almost like one of those cartoons where the witch looks at the baby and the baby shrivels. I didn’t want my perfect baby to have Thatcher’s eyes upon him.” She added, more persuasively, that Thatcher “was not supportive of other women”.

While there can be no doubt that any of the three women in Labour’s four-person leadership contest would ever knowingly shrivel a baby, the current leadership election has renewed arguments about the significance of a party leader’s sex. Emily Thornberry, who previously supported first Miliband, then Jeremy Corbyn over female candidates, says the party must now pick a woman – her – because she would frighten Boris Johnson. Having earlier supported Miliband, Burnham, then Corbyn, her colleague, Lavery, now the party’s chairman, says: “We need a female leader of the Labour party. Stand aside, Keir.”

Lavery’s choice is Rebecca Long-Bailey, partly, he explains, the better to soothe anxious comrades, because “she isn’t frightened of anybody – she’s not even frightened of me”. Yet more impressively, Long-Bailey obviously isn’t bothered by Lavery’s genuinely scary assistant, Paul Robertson, unsacked after tweeting that Theresa May would “look better with a noose round her neck”.

As instructive as it is to be re-educated in sex equality by Lavery, his historic reluctance to tell male candidates to “stand aside” is surely an admission that, notwithstanding the asset that is femaleness, other leadership skills such as actual policies, demonstrable competence and an attractive personality can be yet more important. If not, why not concede that Thatcher, being female, got one thing right?

For Lavery, other priorities might include not being scared of Lavery, and awarding Corbyn 10 out of 10 for perfection. His fellow feminists may, however, attach greater importance to a candidate’s achievements for women. Without, for a second, dismissing any candidate as an aberration, their significance, as women, is doubtless meant to be more than physically verifiable; more even than a gendered knack of looking kindly around babies. Can they cite any contributions towards, say, reducing violence against women and girls, protecting women’s rights, enhancing women’s autonomy?

Can they cite contributions towards, say, reducing violence against women and girls, or protecting women’s rights?

Apologies if I’ve missed something – although, given the detail supplied with other credentials, eg, familiarity with penury, anything notable would probably have surfaced by now. Thornberry? For a person who volunteers that her childhood cats had to be put down, presumably after the family was reduced to competing with them for mice, this infinitely adaptable MP has to date offered little by way of hard-won, womanly insights beyond a new claim, arguably more offensive than Thatcher’s household hints, that “any mother” would agree she was right to nick selective state school places (in a different county) intended for the less privileged.

Though there is still time, of course, for this formerly doting apologist to discover a confidential memo in which she beseeches Corbyn not to mutter “stupid woman” under his breath.

Long-Bailey? To date, the Corbynist idea of charisma appears to have achieved less for women than, despite not being a woman, did her main rival Keir Starmer, as director of public prosecutions. When he left, in 2013, Thornberry said (of the person she now calls a “machine politician”): “His values shone through with his emphasis on fighting violence against women and girls. He fulfilled a difficult role in a progressive and compassionate way… he has done his utmost to transform the CPS’s record on rape and domestic violence, delivering improved conviction rates for both.”

Starmer, moreover, supports women’s reproductive autonomy. Long-Bailey favours restrictions on abortion (and likewise opposes assisted dying). Both politicians having chosen, like Thornberry, to serve Corbyn, they are contaminated, as Lisa Nandy is not, by the party’s documented misogyny; its ineffectual approach to sexual misconduct. Actually, whatever did happen to the open letter, signed by more than 100 former and current staffers, about the handling of allegations against David Prescott, one of Corbyn’s senior aides, condemning “an environment where sexual harassment and bullying are not taken seriously”?

And so to Nandy. Unimpeachably female. Unapologetically at ease with Brexit, and therefore with an outcome that is authoritatively predicted to hit women, via eroded legal protections and increased exposure to poverty, harder than it will men.

Stand aside Keir? Why? What explains this prodigiously late-onset Labour reverence for female iconography? True, we can’t be absolutely sure Starmer wouldn’t shrivel a baby at 100 paces. But given the available competition, that could be the most compelling argument against him.

• Catherine Bennett is an Observer columnist