“My school, we affectionately nicknamed it Avonjail, but it was called Avondale, Avondale high school in Stockport. I left with no GCSEs above a D,” says Angela Rayner, with a school rebel kind of a grin, as we start our interview. Brought up on a Stockport housing estate by her mother, who could neither read nor write, Rayner was pregnant when she quit full-time education. “I kind of left at 15.”

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She is now 39 and as shadow education secretary could soon be in a Labour cabinet. After a 45-minute discussion in which we cover education policy, inequality, Brexit, the state of the Labour party and Boris Johnson at breakneck speed it seems natural to wind up by asking what ambitions she could possibly have left. Would she like to complete her extraordinary journey by, one day, becoming Labour leader?

She is prepared for the question and has her politician’s answers ready. “I am quite happy with the one we’ve got,” is the first reply. But when pressed about what might come to pass in a post-Jeremy Corbyn era, she accepts, based no doubt on personal experience, that anything must be possible. “Never say never,” she says with a big wink.

For now, Rayner is focused on trying to reshape the education system so people from working-class backgrounds like hers can scale the heights. Paradoxically she shines out as an embodiment of social mobility in a system she believes is designed to choke it off. Modestly she puts her own ascent down to good fortune. “I don’t want to be the lucky one that was plucked from my background to get here,” she says.

“I actually want it to be fundamentally part of what we do, making sure there is justice across the board so that the opportunities I have are available to all my friends, and not just for the lucky one to get that advancement.”

Her approach to education policy is driven by frustration that the school system abandons so many young people. “I am not OK with the system that allows certain people to fail or be chucked out. I don’t accept that.”

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She quickly reveals a big reform plan to be announced at the Labour conference, which begins in Brighton this weekend, that she believes will help address some of the unfairnesses that rankle with her. A Labour government with her as education secretary would, she says, abolish the schools inspectorate Ofsted because in her view it creates more inequality and makes the lives of many teachers a misery. “I believe Ofsted measures poverty,” she says. “It measures deprivation. It doesn’t measure excellence.”

There are too many cases, she believes, where Ofsted’s judgments reflect the affluence of a school’s intake and the social class of its pupils rather than the performance of the school. In her view, it encourages competition of the worst sort which in turn creates such high workloads and stress among teachers that many decide to leave for other jobs. “The whole system at the moment is a hothouse to pitch school against school.”

So under Labour all schools and education providers would be subject to regular “health checks” led by local government. Inspectors would then follow up on concerns raised by the checks, and by parents, teachers and governors. “This means that in-depth inspections will arise from a genuine need, instead of taking place at random,” she says.

It is a radical move, and follows previous commitments she has made to halt the expansion of academies and scrap free schools. Too many of the latter have been set up, she says, in middle-class areas where they are not really needed – with the result that resources have been diverted from those in greatest need.

There are limits beyond which she will not go, however, at least not yet. Rayner is no firebrand leftwinger, not even a dyed-in-the-wool Corbynista. She got into trouble with many on the left this year for daring to praise Tony Blair after he gave an interview about a second referendum. Some believe she occupies an interesting space somewhere between Corbyn’s Labour and Blair’s New Labour.

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This week in Brighton delegates are expected to discuss a motion to “Abolish Eton” from the campaign group Labour Against Private Schools, which calls for the removal of independent schools’ charitable status and would force them to let state schools use their assets. While the shadow chancellor, John McDonnell, has backed the motion, Rayner is more cautious, saying it is up to conference and the party to debate it. She is clearly in favour of ending charitable status if at all possible.

“Private schools are gaming the system,” she says. “There is way too much state money going in, and people who go to private schools seem to be given a head start for all of the top jobs and that’s something that needs to be dealt with as well.” But if more draconian measures are in the pipeline she is not ready to reveal them now. Beforehand there is much serious debate to be had.

On Brexit, she denies the party’s position is “fudged” – but readily admits it is has been difficult to reach a position around which the party can unite. Like the Labour leadership she prefers to talk about anything but Brexit. She seems genuinely enthused, however, about the prospect of a general election in which she wants fairness to be Labour’s guiding light. Intriguingly she suggests opportunity for all was what guided New Labour when she was growing up and helped her get to where she is today.

“I think that most people when they are faced with more austerity, more cuts, a more unequal society or one that’s genuinely full of working-class people like me want to make sure that their kids get the opportunities that the last Labour government gave me. I think we will win hands down.”