President Trump, 77, says he supports mental competency tests for holding political office although he doubts the constitutionality of requiring them. Expect the topic to gain traction as partisans highlight his and Mr. Biden’s stumbles as evidence that they’re not up to serving as commander-in-chief.

On Sunday, Mr. Trump told the host of “Meet the Press,” Kristen Welker, that he “aced” questions administered at Walter Reed National Military Medical Center in 2020. “I got everything right,” he said. “I’m all for the tests.”

In February, Governor Haley, 51 and a candidate for the GOP presidential nomination, proposed mandating a test for politicians over 75. The following month, First Lady Jill Biden called the proposal “ridiculous” on CNN. “We would never even discuss something like that,” she said of Mr. Biden.

Mr. Trump said that “testing would be a good thing … to make sure everyone is just fine, but a lot of people say that can’t happen because of the Constitution,” which requires only that a president be 35 or older, a natural born citizen, and American resident for at least 14 years.

The 14th Amendment — ratified in 1868 after the Civil War and floated as disqualifying Mr. Trump over January 6 — added limits. Those who “engaged in insurrection or rebellion against” America or gave “aid or comfort” to its enemies are ineligible unless granted amnesty by a two-thirds vote in Congress.

In 1995, the Supreme Court ruled in U.S. Term Limits v. Thornton that states cannot impose additional restrictions, such as California requiring candidates to release their tax returns. So, short of another constitutional amendment, cognitive tests will remain voluntary, a political issue rather than a legal or medical one.

During his annual physical last November, Mr. Biden chose not to undergo the test as Mr. Trump had done. Silent on the question, he bristles when asked about his age and has a stock response to those who question his fitness: “Watch me.”

The American people already keep an eye on modern presidents in a way they couldn’t, say, President William Henry Harrison. Sixty-eight when he assumed office in 1841, he remained the oldest chief executive until in 1981 President Reagan took the oath of office 16 days short of his 70th birthday.

Harrison and Reagan were both accused of having lost their edge. Less well known were the problems of President Lyndon Johnson. Betty Boyd Caroli, author of “Lady Bird & Lyndon,” told me in our History Author Show interview that today, LBJ would be diagnosed as manic depressive or bipolar. 

Presidents John Quincy Adams, Lincoln, Pierce, and Theodore Roosevelt all dealt with depression, though none are known to have taken to bed unmoving the way LBJ did, always having to be coaxed back to work by his wife.

A Democratic candidate in 1972 for vice president, Senator Eagleton, 42, was kicked off the ticket when it was learned he had been treated for depression. A test might have disqualified him, too. Winston Churchill battled what he called “the Black Dog” his entire life and treated it by one of its greatest cures, painting.

Asked about the question of age, Mr. Trump said he isn’t “anywhere very near 80” but cited examples of people much older who retain sharp minds. “I don’t think Biden’s too old,” he said, “but I think he’s incompetent, and that’s a bigger problem.”

Cognitive tests will always be subjective anyway, unlike ones for HIV or Leukemia which give clear positive or negative results. The Constitution empowers a free press to inform the public about a candidate’s fitness for office, no matter how much citizens might prefer a shortcut. 

So it will be up to candidates to decide whether they undergo mental checkups, and on Election Day, the people will cast ballots based on job performance, personality, and policies. Responses to an arbitrary set of questions may play a role, but no test will ever be able to choose a good president.