Peter died in 67 AD! Of course he disappears; however, a new pope (pope, btw, means 'papa') was elected. Perhaps you neglected to read the information linked above The First 10 Popes of the Catholic Church, with documented references.
To then conclude he was used as more than a foundation stone, as Scripture says, is where your assumptions lead you to invent material to fill in the historical gaps between what is written and what you believe.
No invented material here, nor was Peter a "foundation stone". Peter’s preeminent position among the apostles was symbolized at the very beginning of his relationship with Christ. At their first meeting, Christ told Simon that his name would thereafter be Peter, which translates as "Rock" (John 1:42). The startling thing was that—aside from the single time that Abraham is called a "rock" (Hebrew: Tsur; Aramaic: Kepha) in Isaiah 51:1-2—in the Old Testament only God was called a rock. The word rock was not used as a proper name in the ancient world. If you were to turn to a companion and say, "From now on your name is Asparagus," people would wonder: Why Asparagus? What is the meaning of it? What does it signify? Indeed, why call Simon the fisherman "Rock"? Christ was not given to meaningless gestures, and neither were the Jews as a whole when it came to names. Giving a new name meant that the status of the person was changed, as when Abram’s name was changed to Abraham (Gen.17:5), Jacob’s to Israel (Gen. 32:28), Eliakim’s to Joakim (2 Kgs. 23:34), or the names of the four Hebrew youths—Daniel, Hananiah, Mishael, and Azariah to Belteshazzar, Shadrach, Meshach, and Abednego (Dan. 1:6-7). But no Jew had ever been called "Rock." The Jews would give other names taken from nature, such as Deborah ("bee," Gen. 35:8), and Rachel ("ewe," Gen. 29:16), but never "Rock." In the New Testament James and John were nicknamed Boanerges, meaning "Sons of Thunder," by Christ, but that was never regularly used in place of their original names, and it certainly was not given as a new name. But in the case of Simon-bar-Jonah, his new name Kephas (Greek: Petros) definitely replaced the old.
Now take a closer look at the key verse: "You are Peter, and on this rock I will build my Church" (Matt. 16:18). Since Simon’s new name of Peter itself means rock, the sentence could be rewritten as: "You are Rock and upon this rock I will build my Church." From the grammatical point of view, the phrase "this rock" must relate back to the closest noun.
1. Your Greek is lacking - both vocabulary and grammar.
2. Right after calling Peter a small moveable stone, Christ addresses Peter as Satan. Using your back-fill logic, no one else in the Bible has ever had their name changed to Satan except Peter- changing his status and symbolizing his new name.