The way God brings the Church together isn't always the same. The reasons for that are known only to God, but this is a pretty good example of that. For Peter and Cornelius, he came through visions and dreams. He didn't do that for the entire church, because it's a partnership that God has made with man. In this case, he let Peter take the stage. Of course, the Spirit is in all of it, but it's very humbling to think that God allows us to play important roles in his plan.