Wow. You’re all over the map. In your absurd etymology, all names that start with P and end in R come from a common root.
Peter != Pater. Peter is derived from the Greek for, not father, but rock — petroleum is oil drawn from rock. Petrograd is a city built from rock. Wood that turns into rock is petrified. When Jesus clapped Peter on the shoulder and said “on this rock I shall build my church,” he was making a pun.
Jesus never gets enough credit for having a sense of humor.
Pater is not greek, but Latin — before Vatican II, Catholics didn’t say “Our Fathers,” They said “Pater Nosters.”Peter and Pater share a number of letters in common, but they do not share a common derivation.
But "pater" is Greek and the Greek word for "father", as well as the Latin word.
My question would be how did the word "Petros" in Greek, which was "Petrus" in Latin, evolve into "Pietro" in Italian and then "Peter" in English???? Wouldn't theologians want to carry over a person's name from one language to another as close as possible to the original language. It's done with "Cephas" for Peter. Then why not "Petros"???
It is conceivable that as the Roman Church began to view "Petros, Petrus" as the Pater [Father] of their Church, that "Pater" would have been Petrus's title: "Pater Petrus" [Father Peter] -- shortened in time to just the Italian "Pietro" and later "Peter", a name remarkably similar to "Pater" in both spelling and meaning to those in the RCC -- Peter, the Pater of their Church.