The Labour leadership candidate Jess Phillips says she believes the public is prepared to pay more tax to get an NHS-style social care system that looks after elderly and vulnerable people.

Phillips, whose campaign slogan is “speak truth, win power”, said she stands ready to work with the Conservatives and other parties to solve a problem that has dogged successive governments.

She said a “citizens’ assembly”, similar to the Irish model that proposed new abortion laws, could allow ordinary people to work out how a better system could be funded.

Labour leadership: Long Bailey, Nandy and Phillips win backing Read more

“The public are not stupid and they are not greedy either and they know that a service has to be paid for,” she said. “I’m the kind of leader who would try to have honest and difficult conversations.”

Arguing that politics needed to be taken out of the equation, she said all the parties needed to come to a consensus, after having collectively “failed every family with a disabled dependent adult or elderly person with our political ping-pong on this matter”.

“We’ve got to be brave. The NHS was hard to deliver, so was the minimum wage. It’s time now – we need to have a proper conversation about how much is the individual cost, how much is the burden that we’re all going to share together, and how much are we going to put on older adults now versus a future system like national insurance,” she said.

Phillips said she would not want to prescribe how people should contribute more as the public should be trusted to come up with a solution through a “citizens’ assembly to make recommendations on how to deliver and pay for social care”.

But she added: “If you were to ask somebody in my constituency, what do you think we should do? They’d say: ‘I think it should be free. Everybody should be looked after.’ Do you think you should lose your house? The answer would be no. Do you think that young taxpayers with masses of debt should pay for it? ‘Not necessarily.’ Do you think everyone should pay for it? ‘Yes.’ In this conversation, there is an answer. And I think we can get there.”

She highlighted how the burden of care often fell on women in families, and spoke of giving up her master’s degree to care for her mother, who died from cancer.

We’ve got to be brave. The NHS was hard to deliver, so was the minimum wage. It’s time now – we need to have a proper conversation.

Phillips made it through the first round of the leadership contest on Thursday with nominations from 22 MPs, joining the favourites, Keir Starmer and Rebecca Long Bailey, as well as Lisa Nandy. She will now have to work to get the support of at least 33 constituency Labour parties, or three affiliates including two trade unions making up at least 5% of affiliate membership.

The 38-year-old, known for her outspokenness and for campaigning on women’s rights, acknowledged that passing the next round of the contest would be “a challenge” but said she was relishing the chance to talk to members, whether or not they agreed with her.

She also insisted it would not be as hard as people thought to bring Labour back together after its disappointing election result, divisions over Jeremy Corbyn’s leadership and the party’s problems with antisemitism.

Facebook Twitter Pinterest Jess Phillips and Keir Starmer on the BBC’s Andrew Marr Show. ‘For lots of people, I don’t look like any prime minister that has ever existed,’ she says, adding that it is time for Labour’s next leader to be a woman. Photograph: Jeff Overs/BBC/PA

“We have a set of ground rules and they have to be followed, but we’re going to need a massive overhaul of what our expectations of members are and what members’ expectations of the party are and, when there is a rule, it has to be followed no matter who you are,” she said.

“And you also do it with love. Has nobody come from a family that had arseholes in it? And people who you fell out with? I refuse to believe this rhetoric that the Labour party can’t get under one big umbrella with a common enemy – sometimes a common enemy is an absolutely delightful unifier.”

Her critics have sought to paint her as not working class enough because her mother worked her way to a senior position in the NHS and too rightwing because she has criticised Corbyn over antisemitism and women’s representation. But Phillips dismisses this as “online propaganda” and argues that most of the vitriol against her is only to be found on the internet, while Labour party meetings are mostly “all about samosas and talking about socialism”.

The greatest lie that was ever told is that I’m some sort of rightwinger.

Claiming to have talked round two fans of the leftwing Novara Media site to her cause at a dinner last week, she said: “The greatest lie that was ever told is that I’m some sort of rightwinger.”

She highlights her work running a women’s refuge, her childhood going to “women’s liberation playgroup” and her father being someone “who makes Jeremy Corbyn look like Tony Blair” as evidence of her deep Labour roots.

But she adds: “I am totally realistic. There are people in the Labour party who literally seem to viscerally hate me. I’m not underestimating that. But there are people in the Labour party who viscerally love me. It’s sort of Corbynite, isn’t it?”

With her Birmingham accent, tattoos and casual turn of phrase, Phillips highlights how different she would be from any prime minister who had gone before her but argues that Labour now needs to make a bold choice.

“I recognise that for lots of people, I don’t look like any prime minister that has ever existed. I get that,” she said. “But for the Labour party to look at me versus some of the other candidates and say, ‘She doesn’t look prime ministerial’, there’s some obvious issues in there that the Labour party has with regard to all the other previous leaders.

“Do you remember a prime minister with a regional accent? Do you remember a prime minister who genuinely sounded like the people in the country when they spoke? I am the bold choice to put up against Boris Johnson and it’s going to take bold to beat Boris Johnson.”

We’ve been at the forefront of fighting for women’s rights and we always seem to just fall at this hurdle.

With Keir Starmer now the favourite in the competition, having got the biggest chunk of MP nominations and the backing of Unison, Phillips also made the case that it was time for the next Labour leader to be a woman.

“We’ve been at the forefront of fighting for women’s rights and we always seem to just fall at this hurdle,” she said.

Phillips conceded it was a big step up from being a backbencher but said she was “as ready as much as anyone’s ever ready” for the task.

“Anyone who says they’re not daunted by it is arrogant, doesn’t deserve the job and doesn’t understand the weight of responsibility”.

As for putting together a shadow cabinet, Phillips said she would have a broad range of people from across the party who had backed different candidates.

“I don’t give a toss actually which faction people have been in, because I’ve been put in one that I’m not in,” she said, referring again to the idea that she is on the right of the party. ‘“I’ve never been part of any of these Labour groups, that everybody’s part of. I’m from the Labour party.”