Wes Streeting has insisted he can win over the Labour left, as he launches a shadow campaign for the party leadership, saying he has “beaten the odds” throughout his life and can do so again.

The former health secretary, who called on Keir Starmer to resign as he quit the cabinet last week, warned Labour MPs that drifting on with Starmer in charge risked a Joe Biden situation that would usher in a Reform government.

Although there is no official leadership contest – Starmer is still in Downing Street and the favourite, Andy Burnham, is campaigning to win a byelection – Streeting said it was “inevitable” the prime minister would stand down.

In an interview with the Guardian, the Ilford North MP set out his plans for government, including moving 175,000 children out of temporary accommodation, speeding up the establishment of a national care service, bringing in a wealth tax and “looking again” at plans for refugees.

Starmer’s political authority has been perhaps fatally undermined over the past two weeks after Labour’s devastating election results, with Streeting’s departure from the cabinet and Burnham’s selection to fight the Makerfield seat appearing, in the view of many MPs, to seal his fate.

Many observers were sceptical that Streeting had the support of the 81 MPs required to trigger a leadership contest when he stood down. He insisted he had the necessary numbers but said he had held back because he would have been accused of pulling a “fast one” on Burnham if he had gone ahead before he was able to run.

Streeting denied he was motivated by unbridled ambition for the job, or that he was planning to abandon his run for the leadership and fall in behind Burnham, should he win the byelection, in order to secure a big job.

“That’s not what motivates and drives me at all. I do think it’s inevitable that Keir has to go. I think he’s lost the support of much of his cabinet, lost the support of much of the parliamentary party, and as we saw a fortnight ago, he’s certainly lost the support of the country,” he said.

“I know that I start the race as the underdog, but I’ve been the underdog all my life. I’ve had to beat the odds – from the fact that I ended up in parliament rather than prison like my granddad, that I ended up going from one of the toughest state schools in London to one of the best universities in the country.

“People will say obviously Keir can’t lead us into the next general election. So you’re basically conceding there’s a problem … but you’re saying, let’s go on for just another couple of years of inertia and then change close to the election. That didn’t work out very well for Joe Biden and the Democrats.

“I genuinely think – and I say this with sorrow more than anger – that if we carry on as we are, with a leader that we have, we will hand the keys of Downing Street to Nigel Farage. I do not want that on my conscience.”

Although he trails Burnham in polls of Labour members, and is seen as being on the right of the party, Streeting believes he can still win over the Labour membership, and denied he was suddenly pitching left.

“I think I can win. Yes, I think I can persuade people. The things I’m talking about in this campaign … these are things I’ve talked about throughout my career.”

Streeting’s plan to move 85,000 families out of temporary accommodation, which currently costs £3bn a year, relies on speeding up the delivery of social homes. The government currently plans to build about 300,000 social and affordable homes over the next 10 years, with at least 60% for social rent.

Under his plans, only larger developments – possibly of 2,000 homes or more – would be subject to environmental impact assessments. Developers would have to bid against each other for land that already has planning permission built in, with the value going back to councils to build more social homes and developers able to start building more quickly.

Councils that did not build enough new homes would be stripped of the power to do so, and this would be given to the Planning Inspectorate instead to drive them through.

A Streeting government would speed up the Casey review, which aims to establish a national care service in England but is not due to report until 2028. “I think the timetable is bonkers, to be honest,” he said. “And this was a fight I lost with No 10 and the Treasury. So I’d bring that commission forward.”

He said nobody should have to sell their home to pay for social care, and that the risk and cost should be socialised in the way that it had been for health.

While the government has already brought in a fair pay agreement for care workers, Streeting said he would introduce a national sponsorship body for immigrant workers in the sector, arguing that too many were experiencing “legal modern slavery”, reliant on bad employers for visas.

Burnham has backed Shabana Mahmood’s controversial changes to the immigration system. Streeting also said he broadly supported them, but added there were some “sharp edges” he would want to look at again.

He agreed with the principle of an earned right to indefinite leave to remain – in particular for social care workers – but suggested that giving refugees only temporary protections was something “we need to look at a bit more carefully”.

Streeting said he would stick to Rachel Reeves’s fiscal rules. “You start playing fast and loose with your fiscal rules, your costs of borrowing shoot up. That means less money to invest in public services or back in people’s pockets. That is a total dead end.”

Burnham this week sought to calm markets by committing to the fiscal rules, after previously suggesting they could be changed, for example by exempting rises in defence spending.

The bond markets wanted “fiscal discipline” above all else, Streeting said, suggesting that “shooting our mouths off” on the economy led to higher borrowing costs. Asked if this was a repudiation of Burnham, he said his rival did not “live rent-free in my head”.

He confirmed plans for a wealth tax, by equalising capital gains tax with income tax, said he could look at changing inheritance tax but would “tread carefully”, and suggested he would not increase corporation tax. “You’ve got to keep it competitive,” he added.

Asked how he would deal with the US president if he made it to No 10, he said: “Donald Trump is someone who I think respects straight talkers and respects people who say what they mean and mean what they say. We won’t always agree, but that is how I would approach”.

He trod a careful line on Israel, supporting its right to defend itself in the wake of Hamas’s deadly attacks on 7 October 2023. But he added: “What we’ve seen in terms of the loss of life in Gaza goes well beyond self-defence and is unjustifiable. We should say so.”

He stopped short of describing what was unfurling in Gaza as a genocide, however, saying it was up to international courts to determine that, rather than politicians.

Streeting has urged a “maximalist” position towards closer relations with the EU, sticking to Labour’s red lines this side of an election, but said he would like to see the UK rejoin in future. He denied he was deliberately stirring up the Brexit issue for Burnham, who is fighting in a seat that voted to leave the EU.