Shelters are helped by a burgeoning network of rescue groups. They shuttle dogs from high-kill shelters, usually in the South and Southern California, often to foster homes and adopters in the Northeast and Northwest, where spaying and neutering campaigns have reduced puppy availability. (What Times readers had to say about the tests.)

It is impossible to know how many euthanized dogs scored false positives on behavior testing. Though rare, false negatives also can occur and have proved tragic. In December, workers at Animal Care Centers of New York City saw nothing remarkable on a standard behavior test of a dog named Blue, but noted that he had been surrendered for biting a child. A rescue group retrieved him. Blue eventually wound up in a retraining center in Virginia. On May 31, he was finally adopted; hours later, he attacked and killed a 90-year-old woman.

Some high-volume shelters cannot afford time for evaluations, much less daily walks for dogs; others have begun de-emphasizing their significance. Even Emily Weiss, the A.S.P.C.A. researcher whose behavior assessment is one of the best-known, has stepped away from food-bowl tests, saying that 2016 research showed that programs that omit them “do not experience an increase in bites in the shelter or in adoptive homes.”

Still, Jennifer Abrams, head of the behavior and enrichment staff at Animal Care Centers of New York City, which sees 8,900 dogs a year, said that anxious adopters needed assurances. “People want to know what they’re getting — that a dog won’t bite, yell and scream at other dogs on a leash,” she said.

But predicting an animal’s behavior belies the nature of dogs, Ms. Abrams said: “A dog’s behavior is based on stimuli in the moment.” Ms. Abrams’s team conducts assessments, considering them snapshots, while gathering information throughout the animal’s stay.

In the surge to modernize shelters, tests were an attempt to standardize measurements of a dog’s behavior. But evaluations often became culling tools. With overcrowding a severe problem and euthanasia the starkest solution, shelter workers saw testing as an objective way to make heartbreaking decisions. Testing seemed to offer shelters both a shield from liability and a cloak of moral responsibility.