Wes Streeting and Andy Burnham have criticised Tony Blair’s “striking weakness” in failing to engage with inequality, as senior party figures hit back at the former prime minister’s castigation of the Labour party.

Blair has published a lengthy critique of Labour’s time in office under Keir Starmer, arguing for the government to crack down on welfare spending, abandon restrictions on oil and gas production, and smooth relations with Donald Trump.

It also criticised the policy proposals of Burnham and Streeting – both widely expected to challenge Starmer for the leadership should Burnham win the Makerfield byelection.

In an article for the Guardian, Streeting said that in Blair’s essay, “the defining issue of our age is barely confronted at all. Inequality – the economic, social and democratic fracture running through modern Britain – is treated as peripheral rather than fundamental.”

Burnham, the mayor of Greater Manchester, also said Blair had failed to engage with how inequality was at the heart of Britain’s political issues. “He doesn’t mention inequality once,” Burnham said, adding that he would set out a “considered response” on Thursday.

Starmer was also mulling making his own formal argument in response to the essay, the Guardian understands.

Streeting said: “Inequality, rather than being incidental to the crises reshaping western democracies, is actually their cause.” He said voters’ anger at the disparities was fuelling the growth of populist parties.

“When people believe the rules no longer reward effort fairly, resentment grows. And resentment never remains politically homeless for long,” Streeting said. “The answer to global disruption cannot be a longing for the Britain of the 1970s, nor even the Britain of the 1990s. The task of progressive politics is not to recreate yesterday, but to ensure ordinary working people have power, protection and opportunity in the world now emerging.”

Streeting said Blair was right to praise the opportunities that AI would bring, but the UK must also grapple with the risks it posed to jobs and livelihoods. “It means recognising that economic growth without social justice is ultimately unsustainable.”

He also hit back at Blair’s critique that Labour had hamstrung businesses, saying it was not Labour’s only job to speak the language of the markets better than the Conservatives. “It is to ensure markets serve society rather than dominate it,” he said.

Streeting – who was criticised in Blair’s essay for his proposed wealth tax and for his ambition for the UK to rejoin the EU – said it was vital to “tip the balance of taxation away from work towards wealth”.

And in response to Blair’s suggestions that the government should be prepared to accommodate the US president – including on the Iran war – Streeting criticised Blair’s war in Iraq. “Atlanticism cannot mean automatic subservience,” he said.

“When American presidents flirt with authoritarian leaders, undermine international law or pursue reckless military adventurism, Britain must have the confidence to act independently. We learned at terrible cost in Iraq what happens when loyalty replaces judgment.”

In remarks to the Observer, Burnham said: “[Blair] doesn’t mention inequality once. If you don’t get how that’s driving politics now, if you are not rooting your analysis in the fact that people are unable to live and that things that were taken for granted are no longer affordable, then you are not understanding what’s going on.”

Torsten Bell, the Department for Work and Pensions minister who was a key author of Labour’s last budget, said the former prime minister had made a compelling political argument but one that did not engage in serious policy.

Bell said Blair was right to call out “shallow personality politics”, but “the challenge for the essay is that it doesn’t have a project that remotely fits the time and place we are living in. Saying ‘AI’ is not the same as having a plan for Britain.”

“There is no understanding here of why taxes have risen over the past decade,” Bell added, linking that to higher debt interest costs and the “extremes of austerity for public services”. He said it was “a long way from the truth” that high welfare spending was entirely to blame.

Bell said Blair’s assessment that VAT should have been raised instead of employers’ national insurance was “a recipe for much higher interest rates” and inflation. And there was a “deep inconsistency” in Blair’s approach to the US, “pro-enabling an Iran conflict that has done huge damage to the global economy”.

Blair told broadcasters on Wednesday that he was aiming to start a debate in the party about serious policy, and his advice could not be given in private. “Whether it’s a psychodrama or not, you can debate, but it’s definitely a moment of crisis for the country,” he told LBC.

“And, you know, what I think and hope the essay will contribute to is a debate about what the direction of the country should be, and, I mean, that can’t really be done by private messages. You’ve got to get out there. I don’t particularly want to be back in the headlines again.

“I’m looking at it objectively and thinking, well, we’ve got to change direction, otherwise we’re in real trouble, not because the Labour government is not full of decent people, but because without an understanding of the way the world’s changing and how Britain fits with that change, you can’t really make progress, and you’re going to continue this slide.”