No modern American politician can match Mr. Trump’s record of false or illogical statements, which has invited questions about his intelligence. Insinuations and gaffes have trailed former President George W. Bush, former Vice President Dan Quayle and Joseph R. Biden Jr., now the presumptive Democratic nominee, among many others. But Mr. Trump’s stark pronouncement — on live television, amid a grave public health crisis, and leaving little room for interpretation — was at once in a class of its own and wholly consistent with a reputation for carelessness in speech.

Still, for weeks, the president’s political team has been strikingly explicit about its intended messaging against Mr. Biden: presenting him as a doddering 77-year-old not up to the rigors of the office — and setting off on the kind of whisper campaign that does not bother with whispers…

It is a source of perpetual obsession and manifest insecurity, former aides say, so much so that Mr. Trump has felt the need to allude to his brainpower regularly: tales of his academic credentials at the University of Pennsylvania; his “natural ability” in complicated disciplines; his connection to a “super genius” uncle, an engineer who taught at the Massachusetts Institute of Technology.