There is no evidence that the early church was the catholic church that we have today. On the contrary the Catholic church and the first century church are very different. There is great evidence that the Catholic church did not start until the 300 when the Pope and the offices of the bishopric were set up in Rome.
I address your points using the writings we have available after the Canon of Scripture was closed. These are presented as historical documents, and a summary can be found in "The Teachings of the Church Fathers’ by John Willis, S.J.:
You assert there wasn’t a Pope or “bishopric” (do you mean episcopate?) until 300.
There most certainly was a hierarchical order to bishops.
St. Clement of Rome (d.101 a.d.), Letter to the Corinthians, Ch 42:
“Preaching accordingly, throughout the country and the cities [the Apostles] apointed their first-fruits … to be bishops and deacons of those who should believe”
IBID, Ch. 44:
“Our Apostles also knew, through our Lord Jesus Christ, that there would be contention over the bishop’s office. So, for this cause, having received compelte foreknowledge, they appointed the above-mentioned men, and afterwards gave them a permanent character, so that, as they died, other approved men should succeed to their ministry.”
So we have by the end of the 1st century an episcopate that passes authority from man to man. The authority of the bishops was established over priests and deacons, and the authority of the clergy was established over the laity:
St. Ignatius of Antioch (d. by 117 a.d.), Letter to the Trallians, Ch. 2:
“In the same way all should respect the deacons as they would Jesus Christ, just as they respect the bishop as representing the Father and the priests as the council of God and teh college of the Apostles. Apart from these there is nothing that can be called a Church.”
Letter to the Philadelphians, Ch. 3:
“For, all who belong to God and Jesus Christ are with the bishop. And those, too, will belong to God who have returned, repentant, to the unity of the Church so as to live in accordance with Jesus Christ. Make no mistake, brethren. No one who follows another into schism inherits the kingdom of God (1 Cor. 6:9). No one who follows heretical doctrine is on the side of the passion.”
Letter to the Smyrnaeans, Ch. 8:
“Let all follows the bishop as Jesus Christ did the Father, and the priests, as you would the Apostles. Reverence the deacons as you would the command of God… Wherever the bishop appears, there let the people be; as wherever Jesus Christ is, there is the Catholic Church.”
So there was an episcopate - a college of bishops who worked in community and presided over their specific churches - in place by the end of the 1st century. The papacy - the primacy of the Bishop of Rome and his direct succession from St. Peter - was evidenced shortly thereafter.
St. Irenaeus (d.202 a.d.), Against Heresies, Bk. 3 ch. 3:
“The blessed apostles, then, having founded and built up the Church, committed into the hands of Linus the office of the episcopate. Of this Linus, Paul makes mention in teh Epistles to Timothy. To him succeeded Anacletus; and after him, in the third place from the apostles, Clement was allotted the bishopric. This man, as he had seen the blessed apostles, and had been conversant with them, might be said to have the preaching of the apostles still echoing in his ears and their traditions before his eyes.”
Why Rome? Other than the place where Peter and Paul were martyred, it did not fall into heresy, unlike the church at Corinth (which had trouble even in Paul’s day), as Hegesippus chronicled (though his chronicles remain only in Eusebius of Caesarea’s “Ecclesiastical History”, written two centuries later).
Why Peter as first Pope, or at least the primate bishop? Scripturally, we take this from two sources: Mt.16:16-18, and the first half of Acts. In Matthew, we consider that the Lord spoke to Peter directly, rather than to the Apostles as a whole or the body of all believers. In Acts, Peter has the clear leadership of the Christian community, a leadership that is not contested until Paul begins his ministry some 20 years later.
You may disagree with the Catholic interpretation of these passages, but this has been consistently taught since at least the 2nd century.
Clement of Alexandria (d.216 a.d.), Who is the Rich Man that Shall be Saved? Ch. 21:
After quoting Mt 11:12, “Therefore on hearing those words, the blessed Peter, the chosen, the pre-eminent, the first of the disciples, for whom alone and Himself the Saviour paid tribute, quickly seized and comprehended the saying.”
Tertullian (d. 220 a.d.), On Modesty, Ch. 21:
“If, because the Lord has said to Peter, ‘Upon this rock will I build my church,’ ‘to you have I given the keys of the heavenly kingdom’; or, ‘Whatsoever you shall have bound or loosed in earth, shall be bound or loosed in teh heavens’, you therefore presume that the power of binding and losing has derived to you, tha tis, to every church akin to Peter; what sort of man are you, subverting and wholly changing the manifest intention of the Lord, conferring this personally upon Peter? ‘On you,’ He says, ‘will I build my church’; and, ‘I will give to you the keys,’ not to the church; and, ‘Whatsoever you shall have loosed or bound,’ not what they shall have loosed or bound.”
So the bishops of the Catholic Church were self-referential and self-recognized by the end of the 1st century and the papacy was recognized by the start of the 2rd century.