Your evaluation of that scripture seems legitimate until a deeper examination of the words Christ used shows otherwise. Lets look at what He said.
You are “Petros” (a stone or a rock) and upon this “Petra” (a rock, cliff or ledge - a projecting rock, crag or rocky ground) I will build my church.
There is a difference in the Greek words used, their usage and meaning of the word translated “rock” as can be seen by the following:
Please compare to Isaiah 44:8
Is there a God beside me? yea, no “tsuwr”; I know not any.
TSUWR: rock, cliff. rocky wall, cliff.
PETRA: rock, cliff or ledge.
PETROS: a stone or a rock.
Note also the usage of the word TSUWR in Isaiah 8:14
And he shall be for a sanctuary; but for a stone of stumbling and for a rock (TSUWR) of offence to both the houses of Israel, for a gin and for a snare to the inhabitants of Jerusalem.
This passage is repeated in the NT: Romans 9:33 and 1 Peter 2:8. In both cases the Greek word used is Petra in place of the Hebrew Tsuwr. The same Petra (the “rock, cliff or ledge”) upon which Christ told Peter He would build His church. We can then determine that the rock in Isaiah and the rock in 1 Peter and the rock upon which Christ would build His church are one and the same. This agrees with God's assessment that there is no other rock (TSUWR in Hebrew or PETRA in Greek) beside Him.
Petra is also used to illustrate hearing and doing the words of Christ to the man who built his house on the Petra. Tsuwr is used often in the OT to refer to God as a rock of salvation. All implications of a very large stone capable of being built upon.
Examining the other side of the word for rock, we find that Petros is used exclusively as a name for Peter save for only one time that it is translated “rock” in John 1:42 when Jesus called Peter to be his disciple.
That is the context and content regarding the Petra upon which Christ would build His Church.
It is clear to these protestant eyes that Jesus was by no means calling Peter (Petros) the rock upon which the church was to be built but was actually referring to Himself (Petra). He would otherwise contradict Isaiah 44:8 which states there is no other Rock besides God (translated Petra in Greek). This is echoed by Peter himself when he declared that Petra is the rock that causes men to stumble.
Petra = Jesus Himself, upon which the church is built. No other God besides Petra according to Peter when he quoted Isaiah 44:8.
Petros = Peter, sometimes translated rock according to John.
I understand this objection, but there are many layers within the dialog of Jesus. Before we consider syntax we must consider context. Jesus and the Apostles had the conversation in Matthew 16 in Caesarea Philippi. Located near the Golan Heights the city, previously known as Panis, was built above a huge rock wall also known as the rock of the Gods. At the base of this wall was a flooded cave that local superstitions believed was a passage to the under world. It was before this wall, with the temple to Pan (the Greek God of chaos and confusion, that the conversation took place. When you factor in the possible play on words the syntax is not so clear.
Now, were this the only passage in the Gospels that addressed the primacy of Peter there might be some room for disagreement, but there are about 50 verses that establish it. Peter, like Abram (Abraham) Jacob (Israel) and Sara (Sarah) was given a new name by God upon the bestowing of their holy office. Collectively these leave no doubt.