Apostolic Succession has come down through the Church, not just by laying on of hands, but also by the passing on of the teachings of the Holy Apostles and staying true to how they lived and what they taught. There can be no apostolic succession without keeping the faith.
Also, we are the true church, not just because of our succession, but because of our faith. As is said by some: It is the Church (our Church) that has practiced, taught, fought and died for the faith of the Apostles, and has done so for the last 2,000 years.
The Church is unchanging, yet constantly evolving and growing. It was once a family of several hundred sent out at Pentecost, and has since grown to well over a hundred million here on earth alone. It has been a persecuted, bloodied and beaten family, while also a family with it's share of tiffs and even those who have left the family.
We have our disagreements, but our faith is one, and we are one body. Through the Eucharist, we become the Body of Christ in reality, not in symbology. Each Sunday, we join together as one body with the Church in heaven as a great cloud of witnesses to the world and to worship the Lord.
We have apostolic succession, but if that were all we had, we would be a corpse, a dying group. But because of our succession, and our adherence to the very faith of the Apostles, and all the above-mentioned reasons, we are living, breathing, and ever-growing.
Sorry for such a lengthy reply, but I wanted to really clarify some things according to my little knowledge I've gained thus far. Hopefully this is helpful...