At Mt. Rushmore, Trump Updates ‘American Carnage’ Message for the Election

Ms. Matthews noted that his rhetoric does little more than solidify the voters who were already likely to return to his corner. “He has no interest at all in expanding his base or even pulling back in those who have departed,” she said.

Patrick Murray, the director of the Monmouth University Polling Institute, said that past presidents have typically sought to diffuse cultural battles, “giving people this amorphous kind of middle where they can continue to live.” Mr. Trump, however, is unlike any of his predecessors.

“Donald Trump does not give you that choice — you are either with him or against him,” said Mr. Murray, whose latest survey this week showed Mr. Biden leading 53 percent to 41 percent. “He is forcing people to take sides. And when they take sides, more of them are moving to the other side.”

In Mr. Biden, Mr. Trump also faces a centrist opponent who is not easily branded as a radical liberal, but rather one who is seen as a palatable alternative to some older voters and Republicans in a way that Hillary Clinton was not. Mr. Biden, for instance, has said he does not support defunding the police, and has made careful distinctions between tearing down monuments to the country’s founding fathers and those commemorating Confederate leaders.

That hasn’t stopped the Trump campaign from claiming that in the black-and-white world it wants to present to voters in November, Mr. Biden is on the side of violent looters. “The first instinct of Joe Biden and his party is to agree with the agitators that there is something fundamentally wrong with America and that there always has been,” Mr. Murtaugh said.

The question for Mr. Trump and his political advisers is whether branding Mr. Biden as a puppet for far-left extremists will work. In a statement issued on Saturday in response to Mr. Trump’s speech, Andrew Bates, a Biden campaign spokesman, said: “Joe Biden is running on the opposite values — to win this battle for the soul of our nation, bring the American people together and rebuild the middle class stronger than ever before, bringing everyone along.”

Mr. Trump’s Friday night speech also revealed the president’s concern about his standing with evangelical voters and conservatives, who were crucial to his victory in 2016. On the list of great Americans that he said he wants to erect a statute to honor was the late Supreme Court Justice Antonin Scalia, a conservative favorite.