I noticed that after I re-reviewed. My statement that James was the leader “according to current tradition” is apparently wrong.
According to most of the apocrypha, it does seem to be the majority stance though. Again, who can say what the true answer is?
The Apocrypha, being the key word. 
And?
The Gospel of Matthew was quite likely a later rewrite of the Gospel of the Hebrews. But even if not, the Jewish Christian apocrypha were most likely written at the same time as the orthodox gospels, or at the worst case are based on the pre-Paulian tradition.
But again, regardless of whether Peter or James was the main guy, it’s most likely that New Testament Christianity is an unreliable report of Jesus’ teachings unless you make the assumption that Paul did receive full, magical, instantaneous knowledge of everything Jesus knew after the resurrection, and that the New Testament’s compilation and the decision of what was correct and what was heretical was all lead by a magical hand.
I realize that religious faith means trusting that the book is correct, but practically speaking everything known about the 1st century says that the Paulian tradition was just one of many. Paul wasn’t the only person claiming to be the possessor of the true knowledge. His version of events is just the one that flourished, while others were suppressed, destroyed by the Jewish Roman Wars, and yet others continued on until modern day in small communities through the Middle-East. If Saint Thomas had headed to Rome instead of Syria and India, you’d have a much different religion. As fate had it, it was Peter and Paul who went to Rome.