Perhaps this feast could be considered a bit self-serving, but it does serve as a historical reminder of the place of Peter and the place of Rome in that history. Taken separately and out of the context of the feast, we can think of Peter moving from community to community teaching, solving issues, creating structure, and finally arriving in Rome where he is martyred.
We can think about Rome as the seat of the Roman Empire, and the power that entailed.
We can think about the purpose of the chair in Rome, as seat of power and judgment.
We can also ask questions: for instance, did Peter really reflect Roman custom and use a seat to pronounce judgments and make doctrine, like Pilate did with Jesus? It seems to me that we would be more likely to preserve the altar at which he consecrated the bread and wine and less the chair he sat in during the readings.
But that is not really the point of this feast.
Peter's authority to be the steward of the Kingdom, given to him by Jesus, who was given it by the Father, is what we celebrate. The chair is merely a symbol of that authority. That authority does not end with Peter, for he is the Rock upon which the Church is built - not merely the rock on which it was finished. He is our foundation and we accept the responsibility of authority that was given him and given to all upon whom they laid hands, and in perpetuity until the End - all with the laying on of hands, to continue building up the Body. The Bernini reliquary sculpture in the rear apse of St. Peter's shows the chair as lighter than air, held up by the mere finger tips of those supporting it. That is how we should view the chair, as ethereal but solid.
We are then, I guess, certainly self-served by the Servant of the Servant of God.
Additionally, the rock of the Church is also those who in their care for us give us the gift of the Faith. My dad certainly did this for me and my siblings - and I will miss sharing that with him until that day when he, all the saints of my family in heaven, and the angels come to greet me.
When Jesus went into the region of Caesarea Philippi he asked his disciples, “Who do people say that the Son of Man is?” They replied, “Some say John the Baptist, others Elijah, still others Jeremiah or one of the prophets.” He said to them, “But who do you say that I am?” Simon Peter said in reply, “You are the Christ, the Son of the living God.” Jesus said to him in reply, “Blessed are you, Simon son of Jonah. For flesh and blood has not revealed this to you, but my heavenly Father. And so I say to you, you are Peter, and upon this rock I will build my Church, and the gates of the netherworld shall not prevail against it. I will give you the keys to the Kingdom of heaven. Whatever you bind on earth shall be bound in heaven; and whatever you loose on earth shall be loosed in heaven.”
-- Matthew 16:13-19