When he met Mr. Moon in September, Mr. Kim promised to become the first North Korean leader to visit Seoul. But with his negotiations with Washington stalled, the year ended without Mr. Kim visiting as Mr. Moon said he would, further raising doubts about Mr. Moon’s influence over Mr. Kim.

While Mr. Moon remained preoccupied with North Korea, domestic affairs turned against him. The popularity of his governing Democratic Party has plunged as its politicians have been tainted by #MeToo accusations and other scandals. South Koreans in their 20s, a traditional support base for progressives like Mr. Moon, have been fast losing faith in his government because of his inability to create more jobs, according to the polling company Realmeter.

Older, more conservative South Koreans have rallied in central Seoul almost every weekend in recent months, criticizing as dangerous Mr. Moon’s economic policies and his rapprochement efforts with the North.

But Mr. Moon remains convinced that Mr. Trump’s strong desire to achieve something none of his predecessors could accomplish — ending the North Korean nuclear crisis — and Mr. Kim’s desperate need to improve his country’s economy can create a singular opportunity for both Koreas. Mr. Trump and Mr. Kim, he says, can denuclearize the Korean Peninsula and end the tensions there, opening the way for economically integrating the impoverished North, with its cheap labor, and the South, whose slowing economy needs a new source of growth.

“We have never had an opportunity like this since the Korean War ended in an armistice in 1953, and we should not miss it because it will never come again,” Mr. Moon said last month.

Longtime North Korea observers saw the current stalemate coming when Mr. Trump and Mr. Kim ended their Singapore meeting without sorting out key details, especially what is meant by the “complete denuclearization of the Korean Peninsula.” In the past, the North used that term when it argued that it would give up its nuclear weapons only when the United States ceased hostilities, including ending its military presence in South Korea.