Whether or not Peter himself ever ordained a particular bishop has nothing to do with whether or not the See of Rome is a Petrine See or even the Peterine See. It's The Aposotlic See because that's where St. Peter ended his ministry, where he and St. Paul were martyred.The Pope Proclaims: There are Three Petrine Sees, with Equal Authority and with the Keys
Pope Saint Gregory the Great believed that the Blessed Peter had established
three Petrine Sees of equal authority - Rome, Alexandria, Antioch.
This Triptarchy existed prior to the now familiar Pentarchy, and it is connected
with a belief in a Petrine foundation for each of these three major Sees.
Note well what the Pope says here in his letter to Eulogius of Alexandria:
1. The parts where the Pope speaks of Alexandria and Antioch sharing
the keys with Rome
2. The parts where the Pope speaks of the equality of Rome and
Alexandria and Antioch
3. The parts where the Pope says that all three of these Sees form one
See of Peter over which the three bishops preside.
St Gregory I, Pope of Rome, Epistle XL, writing to Pope Eulogius
Patriarch of Alexandria.
"Your most sweet Holiness [Eulogius of Alexandria] has spoken
much in your letter to me about the chair of Saint Peter, Prince
of the apostles, saying that he himself now sits on it in the
persons of his successors.
"And indeed I acknowledge myself to be unworthy, not only in the
dignity of such as preside, but even in the number of such as stand.
But I gladly accepted all that has been said, in that he has spoken to
me about Peter's chair who occupies Peter's chair. …And to him it is
said by the voice of the Truth, To thee I will give the keys of the
kingdom of heaven (Matth. xvi. 19). And again it is said to him, And
when thou art converted, strengthen thy brethren (xxii. 32). And once
more, Simon, son of Jonas, lovest thou Me? Feed my sheep (Joh. xxi.
Wherefore though there are many apostles, yet with regard to the
principality itself the See of the Prince of the apostles alone has
grown strong in authority, which in three places is the See of one.
For he himself [Peter] exalted the See in which he deigned even to
rest and end the present life [Rome]. He himself adorned the See to
which he sent his disciple as evangelist [Alexandria]. He himself
established the See in which, though he was to leave it, he sat for
seven years [Antioch]. Since then it is the See of one, and one See,
over which by Divine authority three bishops now preside, whatever
good I hear of you, this I impute to myself.”
(Book VII, Epistle XL)