Mr. Trump himself was not displeased with his own performance, according to his advisers. On the contrary, he was elated about the debate and saw it as a successful outing for him, according to three people close to the campaign. Some of Mr. Trump’s advisers, who shy away from giving him bad news, made no attempt to disabuse the president of that assessment…

Before the debate, advisers had tried to prepare Mr. Trump for a question about white supremacy, pointing out to him that Mr. Biden had made fighting racist violence of the kind that broke out in Charlottesville a central theme of his candidacy. Those efforts at preparation did not pay off, and some Trump advisers were privately candid that his hectoring performance recalled how he handled briefings with reporters about the coronavirus last spring, to his political detriment…

“If the ‘suburban housewife’ he keeps talking about really is the whole deal, it’s hard to think he didn’t go backwards with her,” said former Gov. Bill Haslam of Tennessee.

Republicans conceded that it would not be easy to rein in Mr. Trump, whose approach to politics is largely driven by his personal instincts and grievances — and a profound aversion to criticizing anyone whom he counts among his admirers, whether that is President Vladimir V. Putin of Russia, the former Senate candidate Roy S. Moore of Alabama or followers of the online QAnon conspiracy theory.

Other Republicans were despairing not just for their party and its prospects in November but for the country after a debate that they felt represented a low point in American political history.