He instructed Mr. Kerry to keep talking to Mr. Lavrov, and the two came to terms five days later in Geneva. Mr. Obama, his aides said, was determined not to give Mr. Putin a platform to declare Russia was working hand in hand with the United States in Syria, particularly since he did not believe the Russians would abide by the terms of the agreement.

“The president wasn’t prepared to offer the Russians what they wanted most — a symbolic show of U.S. cooperation — until the Russians delivered on their end of the bargain,” said the White House press secretary, Josh Earnest. “That’s why the bargain is structured the way it is. And it’s rooted in our skepticism that they would deliver.”

“The president doesn’t want U.S. credibility to be sullied by Russia’s dishonesty and willingness to sacrifice principle in the name of convenience,” Mr. Earnest added.

Mr. Obama’s skepticism appeared warranted when the aid convoy was hit by a warplane that American officials believe was Russian. White House officials reacted harshly. Benjamin J. Rhodes, the deputy national security adviser, said, “The question is whether or not we just walk away from the table completely at this point, or whether or not we do some more diplomacy and consultation to determine whether or not there is some path forward.”

Again, though, Mr. Obama left it to Mr. Kerry to reproach Mr. Lavrov at a meeting of the United Nations Security Council.

To the extent he mentioned Syria during the General Assembly, it was in broad-brush humanitarian terms. At a meeting with other world leaders on the refugee crisis, Mr. Obama read a letter by a 6-year-old boy from Scarsdale, N.Y., who wrote to him to offer a home to Omran Daqneesh, the 5-year-old Syrian boy from Aleppo who was photographed, dazed and bloodied, after being rescued from an airstrike.

The White House recorded the boy reading the letter aloud, and the video went viral on social media.

Mr. Obama’s struggles with Syria are most palpable when he tries to sum up his foreign-policy legacy. In his speech to the General Assembly, for example, the president cited his diplomatic overtures to Cuba and Myanmar, as well as the Iran nuclear deal and the Paris climate accord, which he said exemplified the power of global collaboration. But when he referred to Syria, Mr. Obama spoke of constraints rather than possibilities.