Where were all his so-called "cousins" who would be obligated to take care of their family member (Mary)? He gave the care of Mary over to John because he was there and Jesus knew he could be trusted. It's really that simple.
Jesus is the rock on which His church is built, not Peter. Then you have James, the brother of Jesus, who became a believer and who was the head of the church in Jerusalem. There were a lot of people gathered in the house at Pentecost as well as HUNDREDS who witnessed the resurrected Christ. Do you think the Holy Spirit should have identified everyone that was there? Take it up with Him!
Peter means “rock”.