There are many leaders in each American political party, the leader of their representatives in the house, the leader of their senators, the leader of the party committee, the leader of the governor’s association, etc. But when a party controls the White House, the president is generally considered the head honcho. Part of that is respect for the office, part of it is practicality (the president has the biggest ability of any one person to message a party’s platform), and part of it is mechanics (every four years at the national convention, the party adopts the platform of their presidential candidate, essentially signing up to work for that person if and when they become president to support enacting that platform. Chuck Schumer is the Majority leader of the Democratic Senate caucus, which makes him very high ranking in the party, but Biden would still “outrank” him, so to speak. Jaime Harrison is the DNC chair, which is largely a fundraising and campaign strategy position vs. a position of power.

If the party holds the Presidency, the President leads the party in every way that matters.