In declaring his candidacy for mayor of New York, former Gov. Andrew M. Cuomo painted a picture of a city in trouble, besieged by crime, homelessness and menace, above and below ground. For all these problems, he blamed a central culprit: the failed leadership of the Democratic Party, of which he has been a fixture for most of his adult life.

As he assailed the party’s progressive wing for calling for the defunding of police departments, and Democrats generally for failing to curb homelessness, it was clear that Mr. Cuomo had a wider audience in mind than voters in the city’s mayoral primary election in June.

In a 17-minute video posted on Saturday, Mr. Cuomo invoked a long list of Democratic presidents — along with one famous Democrat who never sought the White House himself but offered a forceful defense of liberalism at the height of Reagan Republicanism.

“F.D.R., John Kennedy, L.B.J., Mario Cuomo, Bill Clinton, Barack Obama taught us what it meant to have a true progressive government: It wasn’t about rhetoric, but results,” Mr. Cuomo said. “They focused on issues that mattered to people in their day-to-day lives, issues that were relevant to them, and then they actually made life better for people. And that is what Democrats must do once again.”

Offering himself up as the latest in a line of long-admired leaders of the party, including his father the three-term governor, is part of Mr. Cuomo’s effort to redeem his own reputation. His standing was battered by accusations of sexual harassment that forced him from office and by questions about his management of New York State during the coronavirus pandemic.

The personal hurdles that Mr. Cuomo faces might prove to be insurmountable: The state attorney general concluded that he sexually harassed nearly a dozen women. New Yorkers could deny him a second chance.

But as he kicked off his bid, the digs against the leadership of his party amounted to an effort to insert himself into a more abstract but nonetheless urgent debate over the future of the national Democratic Party.

It’s a battle that has been bubbling beneath the surface since Donald J. Trump won the White House last fall and Democrats found themselves powerless in Washington.

In his opening video, Mr. Cuomo addressed the uncertain political climate and suggested that the country look to its largest city — and his leadership — for answers.

“At this time, when the nation is searching for its soul, divided as never before, questioning our democratic values, questioning the very role of government and the balance of power, New York must show the way forward and remind the nation of who we are,” Mr. Cuomo said in the video.

“New York says that we can’t run from each other,” he added, “but rather we must turn towards each other.”

Mr. Cuomo, 67, has never been short of ego or ambition. Before he was derailed by scandal, he planned on seeking a fourth term as governor, an achievement that had eluded his father, and he faced perennial questions about his oft–denied presidential ambitions.

While his campaign is new, Mr. Cuomo’s message about failed progressive leadership is not. He spent much of his third term as governor casting his party’s energized left flank as a political foil — when he wasn’t trying to claim credit for its gains. “I am the left,” he said in 2019, after attacking progressive lawmakers in his state.

In the new video, Mr. Cuomo’s critique focused on what he said was the Democratic Party’s failure to solve people’s most pressing problems — affordability and public safety chief among them.

“The founding premise of a progressive Democratic Party is all about serving working men and women,” he said, “but the cruel irony is they are the ones now paying the highest price for New York’s failed Democratic leadership.”

A victory in the New York mayor’s race would probably catapult Mr. Cuomo back to national prominence. Some of his supporters say they could see him even using it as a springboard to national office.

Other Democrats find that laughable. They highlight Mr. Cuomo’s selective retelling of his own political history, which omitted the investigations of his administration’s apparent efforts to conceal Covid deaths at nursing homes and his elimination as governor of cash bail for many crimes, which some Democrats argue contributed to a rise in crime.

“The idea that he is the competent adult in the room who will run on center-left policies — there’s a lot to be desired there,” said Lauren Hitt, a spokeswoman for New Yorkers for Better Leadership, a super PAC set up to oppose Mr. Cuomo. “He offers a template for politicians trying to overcome scandal, but I don’t think a party in search of a message can look to Andrew Cuomo.”

Mr. Cuomo also offered himself up as an experienced executive who could show New Yorkers that government “can actually work and get things done — big, hard, important things — and get them done well.”

It’s an argument that echoes the ambition, if not the delivery, of former President Joseph R. Biden Jr., who hoped to restore faith in government by enacting policies that improved Americans’ lives.

Of course, during Mr. Biden’s tenure, Democrats were able to maintain a delicate peace between their centrist and left-wing factions for the sake of passing an ambitious legislative agenda. Now that they are locked out of federal power, everything from their policy positions to their political strategy is suddenly up for debate.

In addressing policing, housing prices and homelessness, Mr. Cuomo sided decisively with Democratic moderates who would like the party to emphasize law and order.

“To me, the founding premise of a progressive Democratic Party is all about serving working men and women, but the cruel irony is they are the ones now paying the highest price for New York’s failed Democratic leadership,” he said. “This is not progressive policy, but in fact regressive policy.”

As the new Trump administration pushes a right-wing agenda at head-spinning speed, Democrats have quarreled over how aggressively they should fight back. While some have argued to pick their shots or even, as the strategist James Carville suggested, to “roll over and play dead,” younger, more liberal lawmakers are pushing the party to oppose the president’s entire agenda.

“A lot of Democrats think maybe you should fight every third day, you should reserve your power and jump out of the bushes at the right moment,” Senator Chris Murphy, the Connecticut Democrat who has emerged as a leading exponent of fierce Democratic pushback, said Sunday on CNN’s “State of the Union,” mocking that cautious approach. “We have to be on the offensive 24-7.”

Mr. Cuomo, by contrast, has notably declined to criticize Mr. Trump — a “conspicuous silence,” according to State Senator Zellnor Myrie, one of his opponents, that is notably different from the tough tone he took against the president during the pandemic. In his video, he struck a note almost of collegiality.

“I will cooperate and collaborate on any and every level,” he said in his announcement video. “I have worked with President Trump in many different situations. And I hope President Trump remembers his hometown, and works with us to make it better.”

When it comes to New York City, Mr. Cuomo seemed to hope that the president would let bygones be bygones to form a new relationship.

He is clearly hoping that the city’s voters will do the same.