I do not see any 'leadership' from Peter.
He does do some talking in this passage.
7 After much discussion, Peter got up and addressed them:
12 The whole assembly became silent as they listened to Barnabas and Paul telling about the signs and wonders God had done among the Gentiles through them. 13 When they finished, James spoke up.
That is because you do not understand how ancient assemblies worked. They spoke in order of precedence. That Peter spoke first shows that he was the leader. He then listens to what others have to say. Throughout the Gospel Peter is always listed first and always speaks on behalf of the rest.