“In the circles that I move, I think there’s almost unanimity of support for him,” Mr. Smith said.

But Mr. Smith acknowledged that in Mr. Biden’s absence, others were making moves in South Carolina. Mr. Smith said that after his defeat in November, he had heard from several contenders, including Senators Cory Booker of New Jersey, Kamala Harris of California and Elizabeth Warren of Massachusetts — none of whom, he said, could count on scooping up Mr. Biden’s network.

The entry of Mr. Biden would immediately upend the still-nascent primary: Ms. Warren joined the race last week and three dozen other Democrats are considering it. Mr. Biden could crowd out other establishment-aligned contenders like former Mayor Mitch Landrieu of New Orleans and former Gov. Terry McAuliffe of Virginia, given that Mr. Biden’s donor base and supporters may overlap with some of their own. And Mr. Biden’s allies are themselves mindful of another Democrat whose blue-collar credentials could pose a challenge to their own coalition: Senator Sherrod Brown of Ohio.

[With Elizabeth Warren visit, Iowans get the first taste of 2020.]

Most of all, a Biden candidacy would also frame the primary around the question of whether Democrats prefer a familiar face and steady continuum of former President Barack Obama’s center-left politics, or a generational change and turn to a more liberal, confrontational approach to Mr. Trump and the right.

Mr. Biden’s supporters argue that Mr. Trump’s incendiary conduct and norm-breaking presidency have so imperiled the country that only somebody with the stature and experience of a two-term vice president can bring back stability. And, they contend, Mr. Trump has diminished the importance of ideological differences within the Democratic Party.

“Variations in policy positions don’t really matter because, to paraphrase the president, we are in a national emergency — but the national emergency is him,” said Lou Frillman, who was a major fund-raising bundler for Mr. Obama’s campaigns and is close to Mr. Biden. “The vice president has all the qualities necessary to win, and I think he has all the qualities the country is going to so desperately need to bring us back from the edge.”

Most of all, Mr. Biden’s allies contend, he would be the Democrat most likely to reclaim the battleground state voters who abandoned the party in 2016.

“If you look at Wisconsin, Michigan and Pennsylvania, the labor folks who voted for Trump, they love him,” former Gov. Jack Markell of Delaware said of Mr. Biden. “He has a connection with these people.”