The Majority Leader does control the schedule. This is because, assuming caucus solidarity, if push came to shove and there was cloture on every step of every bill, that person is essentially controlling the schedule anyways.
Most procedures go along on unanimous consent (nobody objecting) and the floor leader is essentially crafting those agreements out in advance as much as possible. When that person doesn't want something to move, it is usually something that wouldn't move anyways or has been horse-traded for something else to move, etc.
ETA: The Senate has a Republican Majority until Kamala Harris takes office and becomes President of the Senate (and her seat is filled?).