At least you pose a reasonable question..."On what rock, then, is the Church built, if not Peter?"
It is your assumption, not mine, that Jesus has to be referring to a person. Perhaps you believe this because of the "gender" issue, I don't know. But, that Jesus has to be referring to Peter is simply a faulty conclusion reached by the Catholic Church (if you are not Catholic).
My position is that the feminine gender of the noun indicates that He is likely referring to something OTHER than a person. He is likely referring to the statement that "You are the Christ, the Son of the Living God." Upon this statement His Church has been built...not Peter. He could have easily said, "...And upon You I will..." or "And upon the Peter (petros) (pointing to Peter as the Catholics try to maintain)"
But, there is a change in gender. And statements, ideas, would carry a feminine gender NOT because they are girls. Gender doesn't necessarily mean boys and girls. This is basic Greek. Gender differences, however, trigger the recognition of a change of subject. In this case, You are "Peter" (petros) and "This something else" (petra) is what I will build on...not Peter.
The church is not built on Peter. The Church is built on Christ and Him being the Son of God. It is a powerful statement that He is the foundation, the Chief Cornerstone. And, in reality, the Body of Christ actually goes all the way back to anyone rescued since Adam. Abraham is our Father of Faith, is he not? Then if our Church goes all the way back to the beginning of human history and the Roman, Latin, Catholic Church begins with Peter...who's is really older?
Now you seem to be suggesting that Jesus (who IS the Christ) was not a human person?!? That was a common heresy, back in the day, but Christians have agreed on His humanity for over 1500 years now....
But the problem remains -- you've got a feminine gender applied to Jesus who, whatever else He may have been, was undeniably male. Your logic still fails.