Why did Jesus do it? I can come up with a handful of suggestions but they are just that, suggestions, guesses... I try to stick to discussing facts and one fact is that Jesus is the "rock of ages cleft for me".
Jesus is the cornerstone, no Catholic disputes that.
Protestants dissent from the Church that it is in this passage Christ names Peter as the one to lead His church.
There is no question that Peter is called a rock by Christ, and there is no question that Christ then gives him the keys of the kingdom and the authority to bind and loose.
You said it yourself in your reply. Peter is the small rock that comes from the larger rock and because Christ is the larger rock, the bedrock or cornerstone, He has the authority to give to Peter the keys to the kingdom.
No one believes that Peter is anything other than Christ’s representative after His return to heaven. The Church does not teach anything other than that.
God does not change a name without a purpose. Debate over big rock and little rock only obscures the true debate over God’s purpose in changing Simon’s name to Peter(rock).
It makes no sense other than the fact that God’s intended purpose for Peter was that he, Peter(rock)would lead the Church.
now to a person who only knows English this may seem like two different meaning intentions, but that is because English does not have gender assigned to inanimate objects
In other Indo-european languages gender IS assigned to inanimate objects and the Koine greek term for rock is petra, this is not a dimunative
Now, the translator would not give Simeon a girl's name (since Petra is feminine) but would give him the masculine form of Petra which is Petros
And of course Jesus would have spoken this in Aramaic/hebrew. In Aramaic the term for Rock is Cephas which is very clear, Jesus saying "you are Rock and on Rock I will build my Church"
If you want, you can dispute on the precise meaning of this, but the words are clear and it's not Little rock arkansas