Zohran Mamdani, the New York City mayor, said on Sunday that he and a slew of democratic socialist allies who prevailed in recent primary elections are carrying a “national message” to struggling working Americans hungry for a new kind of politics “coast to coast”.

Mamdani made that triumphant clarion call on ABC News’s This Week just five days after he had seen his endorsed candidates win Democratic nominations in three races for New York congressional seats, as well as for five state legislature positions in Albany. He made no effort to disguise his delight that his clean sweep marks a dramatic shift in Democratic politics – not just in New York City, which he has led since January, but also across the US.

He said that collectively they were carrying a “New Deal understanding” of Democratic politics to Congress and on to the “national stage”. It spoke, he said, to Americans feeling exhaustion at struggling to make ends meet “every single day”.

“We don’t have to nationalize that message,” Mamdani remarked. “That is a national message – it’s a national crisis.”

Mamdani’s clear framing of last Tuesday’s victories as a political jolt of nationwide significance poses a challenge to the established leadership of the Democratic party of the sort that they fear most. It is one thing for the mayor of New York to confine his brand of democratic socialism to America’s largest and wealthiest city.

It is quite another when he injects it into Washington.

Established Democrats, branding themselves as “moderates” and Mamdani as “extremist”, have not held back from attacking his burgeoning political movement. Richard Blumenthal, a Democratic senator from Connecticut, predicted: “The effort to nationalize New York is going to fail. What’s happening in New York will be really irrelevant by the time of the [midterm] elections in November.”

Fifteen self-labelled “moderate” Democrats in the US House signed an open letter that, though it did not mention Mamdani or his endorsed allies, was clearly targeted at them. “We are capitalist, not socialist,” they said. “We are mainstream, not extreme. We are proud, not ashamed, of America.”

Asked by This Week what he thought of his critics putting out a manifesto against him, Mamdani joked: “Sounds pretty socialist to me.”

He added: “I’m not interested in writing a manifesto, or frankly, in reading one. I’m interested in delivering.”

Part of the danger that Mamdani represents to establishment Democrats is that he can now substantiate his argument that the party must deliver more to working Americans by pointing to his own concrete achievements. On Thursday, New York’s rent guidelines board voted to freeze rents for about 1m apartments.

“We’ve delivered free childcare for two-year-olds for the first time in New York history,” Mamdani told ABC News, adding for good measure that they had also repaired 165,000 potholes.

He said that such a record showed that democratic socialism was at heart pragmatic – “because if we cannot deliver for working people, then what is it for”?

Mamdani also unleashed piercing criticism of the established leadership of his party. “What we’ve seen over many years is a willingness to not only explain away the status quo, but frankly, even to look to benefit from the status quo,” he said. “And that’s not what working people are looking for from our party.”

He charged the party hierarchy of failing to present a positive vision, and for thinking that merely attacking Donald Trump was enough. “You have got to have something you are not just willing to stand up for – but that you’re also willing to explain how this is relevant to working people,” Mamdani said when asked if one “can’t just be anti-[the president]”.

“I’m leading the wealthiest city in the wealthiest country in the history of the world,” Mamdani said. “But it’s also a city where one in four are living in poverty, and for far too many Americans, those contradictions have become their day-to-day life.”

Tuesday’s primary results in New York were especially poignant because Mamdani’s allies ousted two incumbent Democratic representatives. Brad Lander defeated Dan Goldman, and Darializa Avila Chevalier beat Adriano Espaillat.

Claire Valdez also took the party’s nomination over Antonio Reynoso, the established choice.

Chevalier has attracted particular attention over her political views, which include describing herself as a “prison abolitionist”. In his ABC News interview, Mamdani refused to be drawn into that framing.

He said, “There are prisons. And what we’re also showing in this city is that safety is not something that’s up for debate – it’s something that we’re actually delivering on. And I’m proud to be the mayor of a city that currently has the lowest recorded number of murders and shootings in recorded history in New York City.”

Though several current Democratic politicians are openly spooked by Mamdani’s rapidly growing influence within the party, others are keeping the door open to dialogue. Chris Murphy, another Democratic senator for Connecticut, told NBC’s Meet the Press that the party was a big tent.

He said he did not want to see Democrats embrace socialism. “But we have to understand that people do not believe that this version of capitalism has worked,” Murphy said. “And frankly, it hasn’t.”

Raphael Warnock, the US senator whose fellow Georgia Democrat Jon Ossoff is up for Senate re-election in November, told CNN’s State of the Union: “I don’t believe in abolishing the police. You don’t have a country without borders.”

But he added: “I’m worried about what’s going to happen to ordinary people, who can’t afford their lives right now. And both parties have not adequately responded to this deepening American crisis.”