Greek scholars-even non-Catholic ones-admit, the words petros and petra were synonyms in first century Greek. They meant "small stone" and "large rock" in some ancient Greek poetry, centuries before the time of Christ, but that distinction had disappeared from the language by the time Matthew's Gospel was rendered in Greek. The difference in meaning can only be found in Attic Greek, but the New Testament was written in Koine Greek-an entirely different dialect. In Koine Greek, both petros andpetra simply meant "rock." If Jesus had wanted to call Simon a small stone, the Greek lithos would have been used.
Beyond the grammatical evidence, the structure of the narrative does not allow for a downplaying of Peter's role in the Church. Look at the way Matthew 16:15-19 is structured. After Peter gives a confession about the identity of Jesus, the Lord does the same in return for Peter. Jesus does not say, "Blessed are you, Simon Bar-Jona! For flesh and blood has not revealed this to you, but my Father who is in heaven. And I tell you, you are an insignificant pebble and on this rock I will build my Church. . . . I will give you the keys of the kingdom of heaven." Jesus is giving Peter a three-fold blessing, including the gift of the keys to the kingdom, not undermining his authority. To say that Jesus is downplaying Peter flies in the face of the context. Jesus is installing Peter as a form of chief steward or prime minister under the King of Kings by giving him the keys to the kingdom. As can be seen in Isaiah 22:22, kings in the Old Testament appointed a chief steward to serve under them in a position of great authority to rule over the inhabitants of the kingdom. Jesus quotes almost verbatum from this passage in Isaiah, and so it is clear what he has in mind. He is raising Peter up as a father figure to the household of faith (Is. 22:21), to lead them and guide the flock (John 21:15-17). This authority of the prime minister under the king was passed on from one man to another down through the ages by the giving of the keys, which were worn on the shoulder as a sign of authority. Likewise, the authority of Peter has been passed down for 2000 years by means of the papacy.
There is ample evidence in the New Testament that Peter was first in authority among the apostles. Whenever they were named, Peter headed the list (Matt. 10:1-4, Mark 3:16-19, Luke 6:14-16, Acts 1:13); sometimes the apostles were referred to as "Peter and those who were with him" (Luke 9:32). Peter was the one who generally spoke for the apostles (Matt. 18:21, Mark 8:29, Luke 12:41, John 6:68-69), and he figured in many of the most dramatic scenes (Matt. 14:28-32, Matt. 17:24-27, Mark 10:23-28). On Pentecost it was Peter who first preached to the crowds (Acts 2:14-40), and he worked the first healing in the Church age (Acts 3:6-7). It is Peter's faith that will strengthen his brethren (Luke 22:32) and Peter is given Christ's flock to shepherd (John 21:17). An angel was sent to announce the resurrection to Peter (Mark 16:7), and the risen Christ first appeared to Peter (Luke 24:34). He headed the meeting that elected Matthias to replace Judas (Acts 1:13-26), and he received the first converts (Acts 2:41). He inflicted the first punishment (Acts 5:1-11), and excommunicated the first heretic (Acts 8:18-23). He led the first council in Jerusalem (Acts 15), and announced the first dogmatic decision (Acts 15:7-11). It was to Peter that the revelation came that Gentiles were to be baptized and accepted as Christians (Acts 10:46-48).
For the record, I am a ex JW who is now a Roman Catholic.