Tyler Jones, a South Carolina Democratic strategist who helped flip a Charleston district previously held by Republicans for nearly four decades, worries that having a presidential nominee who backs proposals like free college and single-payer health care would cost the party up and down the ballot.

“If we have a nominee that supports Medicare for all at the top of the ticket, our majority in the House is in serious jeopardy, not to mention a potential majority in the Senate,” said Mr. Jones, who is unaffiliated with a presidential campaign after advising former Representative Beto O’Rourke. “That is not a smart strategy.”

Liberal supporters of Mr. Sanders and Ms. Warren offer a different political analysis of the party’s 2016 loss, saying that it was the failure to energize voters with a bold new vision that cost Democrats the presidency. They argue that polices like Medicare for all, can create a winning coalition by motivating independents who feel alienated by the political system and by energizing Democrats.

On the campaign trail and in debates, Ms. Warren pitches her health care plan by arguing that the current system is failing many Americans, even those who have health coverage, because of high costs. She says she’s asking corporations and the richest Americans to pay more to create a system in which people would no longer face the risk of financial ruin because of medical bills.

While she shies away from offering the kind of explicit political analysis frequently used by her rivals, Ms. Warren argues that the more Democrats talk about what’s broken with the system, the more support for Medicare for all will build. Aides point to polling by the Kaiser Family Foundation showing that a majority of all Democrats and about half of independents favor a national Medicare for all plan.

“We’re not there because we need to be out talking about people’s experiences right now,” she told reporters in New Hampshire.