Still, the show’s relentless focus on political news, and the amount of time Mr. Stewart, who is not granting interviews ahead of his departure, would devote to a certain topic each night, made him an object of fascination, especially among the press.

“I think Stewart has it both ways,” said Jeff Greenfield, the longtime TV news reporter. “He says he’s just a comedian but he’s more than a comedian and I think he knows that. I spent three decades-plus doing network news but if you ask me today, what do I pay more attention to, John Oliver and Jon Stewart or the evening newscast, it’s not close. I get much more out of Oliver and Stewart when he’s cooking than I do out of those formulaic 22-minute newscasts.”

“There are a lot of journalists who watch Stewart and envy the freedom he has,” Mr. Greenfield added. “You can’t go on television when you’re a journalist and say, ‘Senator X is a bald-faced liar.’ ”

The show’s producers and writers knew they had that in their favor. The show’s popularity started to gain steam during the 2000 election, said Madeleine Smithberg, a creator of “The Daily Show,” especially in the monthlong drama after the election that led to the Bush v. Gore decision. “We moved into the forefront in everybody’s mind and everyone was watching us because we were the only ones who could call it for what it actually was,” Ms. Smithberg said.

By that point, Mr. Stewart had been host for two years and was zeroing in on the show’s identity, a significant departure from his predecessor Mr. Kilborn, who was a former sports anchor.

“The show certainly had not found itself by the time Craig had left,” said Lewis Black, the comedian and longtime “Daily Show” contributor who worked with both hosts. “The pieces were in place but it needed somebody to guide the ship in. Jon really became that person.”