Mr. Schumer, 70, has been attempting to channel his party’s sense of impatient purpose: In recent days, he has publicly urged President Biden to “go big and bold” with his economic policies and executive actions, defying pressure from Republicans and a few centrist Democrats to pare back campaign promises.

Over the last week, Mr. Schumer has backed a new push to decriminalize cannabis; signed on to Senator Cory Booker’s Baby Bonds proposal, a plan to address the racial wealth gap; and appeared with Senator Elizabeth Warren and other progressives to call on Mr. Biden to cancel student debt.

On impeachment, too, Mr. Schumer has taken an into-the-breach approach, demanding Mr. Trump’s removal from office the morning after the Jan. 6 attack on the Capitol and casting the upcoming trial as a crucial ritual of accountability even if it is highly improbable that two thirds of the Senate will vote for conviction.