The League always had a rotating membership, and no real leader until the Bronze Age. Then it was Superman, until Zatanna was elected permanent chairperson, I think.
After the Martian Invasion, it was Aquaman until he left, and Martian Manhunter took over until Batman returned after Crisis On Infinite Earths. He remained in charge except when Superman would join the League.
as Mon-El said to Brainiac regarding a new leader (Mon-El resigned),
"it's always about you." - like Brainy, Batman will always have his shadow hovering over the leader, no matter who it is, or who is wearing the Bat costume.