The five men vying to take on Boris Johnson in the Conservative leadership race made a slew of promises to tackle illiteracy, fix the broken social care system and reunify Britain after Brexit as they clashed in the first televised debate.

Johnson was attacked by the foreign secretary, Jeremy Hunt, for failing to appear in Sunday night’s debate, at which he was represented by an empty lectern. “If his team won’t allow him out to debate with five pretty friendly colleagues, how is he going to cope with 27 EU countries? He should be here to answer that very question,” said Hunt.

Much of the leadership contest so far has been dominated by the issue of Brexit, which produced some of the strongest clashes in the Channel 4 debate. But nine years after the Conservatives came to power in coalition with the Liberal Democrats, all five men also used the platform to set out their priorities for renewing their country and their party.

Hunt, who was health secretary for six years, said fixing Britain’s broken social care system was “unfinished business”, and promised to tackle what he called the “national blind spot” of illiteracy, saying: “A quarter of primary school leavers are unable to read or write properly.”

If his team won’t allow him out to debate with five colleagues, how is he going to cope with 27 EU countries? Jeremy Hunt on Boris Johnson

Rory Stewart, who came seventh in the first round of voting last week, also highlighted the state of social care. “It’s a real disgrace: it’s the unfinished revolution of our society,” the international development secretary said. “It is that on which our civilisation should be tested – and it is that which I would make my central priority.”

Michael Gove said he wanted to do more for children in care, and raised concerns about young people leaving university burdened with debt, suggesting the student loans system should be “looked at”.

“It seems to me tragic that when I think of the next generation of children, that they will leave university with debt, they will find it more difficult to have a home of their own; it seems as though their horizons are narrowing just as the world is opening up,” the environment secretary said.

And in an echo of Tony Blair’s “tough on crime, tough on the causes of crime” catchphrase, Gove suggested: “Our approach to knife crime should not just be about law and order – it should be about love and hope.”

Facebook Twitter Pinterest Jeremy Hunt (centre) said Britain’s care system was ‘unfinished business’. Photograph: Tim Anderson/Channel 4/PA

Sajid Javid praised public services, saying: “Like 90% of the population, I relied on public services to get me where I am.” The home secretary also called for more investment in education. Dominic Raab said he would boost degree-level apprenticeships and cut taxes for the lowest-paid.

And despite the public pleas of the chancellor, Philip Hammond, for candidates not to abandon the Tories’ reputation for fiscal prudence, no one mentioned tackling the deficit.

Raab was repeatedly challenged by all four of his rivals for refusing to rule out suspending parliament if he considered it necessary to press ahead with a no-deal Brexit – something he said must be kept on the table. “The minute we telegraph to the EU that we’re not willing to walk away at the end of October, come what may, we lose the best shot at a deal,” the former Brexit secretary said.

But Javid told him: “You don’t deliver democracy by trashing democracy – we’re not choosing a dictator, we’re choosing a PM.” And Hunt said the idea was a “fundamental misreading of what parliament stands for, and what the people in this country would accept”.

Javid, who will need to win over extra support if he is to remain in the race after Tuesday’s second round of voting, painted himself as a fresh face from a different background who could communicate better with non-Tory voters. “It’s important to have someone that can unite people – and my own background, my life experience is very different from Conservative leaders of the past,” he said. “I’m not from central casting.”

Stewart, whose quixotic campaign has involved walking around Britain meeting the public, received a warm response from the studio audience of floating voters, including when he complained that the contest has become a “competition of machismo” over Brexit.

He also took a swipe at Johnson by saying of his rivals in the studio: “I hope it’s one of us who becomes prime minister.”

And after the debate, he won the backing of business minister Margot James, who had previously supported Matt Hancock. She described him as “energetic, determined and embracing the centre ground”.

Margot James (@margot_james_mp) Cheered by the thoughtful and positive #C4Debate @RoryStewartUK had the edge for me, energetic, determined and embracing the centre ground, I will support him for next PM

Stewart claimed he was the candidate who could win over backers of other parties at a future general election.

“I’m not ashamed of the fact that Labour and Lib Dem voters say they would vote for me,” he said. “I think that’s something to be proud of, and I think we need to work to listen to each other, and above all to bring this country together.”

One member of the audience asked the candidates about their greatest weakness. Raab called himself a “restless soul”, Javid said he was too stubborn, and Gove that he was too impatient.

Johnson, who is the overwhelming frontrunner, had declined an invitation to take part in the debate, and is not expected to join a hustings with Westminster journalists on Monday either. But he will join a BBC debate on Tuesday evening, after the second round of voting, when the field will have been slimmed down.

The least popular candidate will be eliminated after Tuesday’s ballot among MPs – as will any contender who does not meet a threshold of 33 supporters, set by the backbench 1922 Committee.

Only Johnson, Hunt and Gove achieved that target last week. Others have spent the weekend trying to win over erstwhile supporters of Hancock, who withdrew after scoring 20 votes in the first round.

Hancock himself, who pitched himself as the right candidate to “turn the page” on Brexit, announced on Sunday night that he was backing Johnson after holding meetings with all the other candidates on Friday and Saturday.

Writing in the Times, Hancock said he felt Johnson had the best chance of uniting the party and the country and would lead as a one-nation prime minister because “that’s how he ran London – consistently – for eight years”. He added: “I have repeatedly argued for a strategy of defeating the danger of [Nigel] Farage by delivering Brexit and defeating the danger of [Jeremy] Corbyn by dominating the centre ground thereafter. That is Boris’s plan and I wholeheartedly endorse it.”

Some moderate Tory MPs had hoped for a “stop Johnson dream ticket”, but talks on joining forces foundered. “Rory, Michael, Saj and Matt all agreed they should band together – but each of them thought it was him that people should unite around,” said one senior Tory.

Further rounds of voting will be held on Wednesday – and on Thursday, if necessary – to weed the field down to just two men, whose names will then go forward to Conservative party members. The result will be announced in the week beginning 21 July.