It is common for Catholics to boast their theory of area Bishops is in all church Fathers after the Apostles, but a careful study will show the opposite to be true. Many times the word Bishop is used is found in the Introductions or footnotes and were added by men hundreds of years later. Clement's writings had been lost for centuries and the footnotes added after the office split. There are occasional quotes with the word Bishop but always used in the sense where bishop and presbyter are used interchangeably. The two terms are not listed as separate offices in a clear way until late in the 2nd century. The earliest Fathers did not make a distinction.
Clement
I Clement is the earliest writer, and he does not use the term bishop but uses the Greek episcopate since the Latin or English wasn't used yet in Rome or Corinth, Episcopos is later translated into the term bishop, Clement used presbyter and Episcopos only. His letter is very damning towards Catholics by listing the necessary qualities of the term Bishop from I Timothy 3, and lists those qualities with the term presbyter in his letter. This shows he saw the two words meaning the same office. They had the same requirements. In Chapter 42 he uses bishops and deacons, leaving out presbyter since he saw them as the same.
"obedient to those who had the rule over you, and giving all fitting honour to the presbyters among you "
"For our sin will not be small, if we eject from the episcopate106 those who have blamelessly and holily fulfilled its duties.107 Blessed are those presbyters who, having finished their course before now,
It is obvious the pffice was episcopate (bishop) while the body or person was termed presbyter or presbytery.
Clement shows multiple rulers over them with multiple presbyters. He also uses the term presbyter in an aged sense, showing the term meant an older man and not a younger priest as Catholics use it. He uses the term episcopate (bishop) in reference to presbyters (elders).
In his list of qualities applied to presbyters, he lists their blameless behavior, being proven, marriages and children, wives , teaching, ruling their own affairs well, all listed in I Timothy 3 for bishops and Titus 1. There is no doubt Clement saw bishop and presbyter as the same office. Clement used episcopos and presbyter as Paul used them in Acts 20. The writer could have known Paul as most speculate since he used them exactly the same.
The idea of presbyters being priests was also interjected into the letter through chapter headings. These came later. The letter itself did not contain christian priests. It spoke of Egyptian priests and Israel's priests and compared them to obeying the episcopate. The chapter heading referred to the priestly office, but the letter itself does not call it that.
Polycarp
Like Clement we have to be careful of the things interjected through the introduction, chapter headings, or footnotes. Polycarp himself never calls himself Bishop of Smyrna, this is in the introduction, footnotes, and headings placed at later dates. The letter itself says Polycarp, with him the presbyters. Polycarp isn't saying they are under him but seems to be giving the idea he wrote, but other presbyters were joining in sending the letter. The Greek "with" can mean location or can mean joint action. In the intro Polycarp is called bishop, but no definite article could mean he was one of several bishops or presbyters.
Like with I Clement it isn't one man taking action but a group, with one writing. If not influenced by the introduction, there is no proof Polycarp was not a fellow presbyter. Since the word bishop and presbyter were used interchangeably by apostles, clement, and Polycarp. As with Clement he doesn't use the idea of Christian priests.
"Wherefore, it is needful to abstain from all these things, being subject to the presbyters and deacons, as unto God and Christ."
Polycarp only uses two offices in the letter, since eriscopate and presbytery were the same. Polycarp nowhere uses the word bishop or episcopate in his letters except for Christ.
Mathetes
Mathetes does not mention bishops, elders, or deacons.
Ignatius
It is obvious that Ignatius is the first to refer to the Bishop and presbyters as separate offices.
"neither do ye anything without the Bishop and presbyters"
"Now it becomes you also not to despise the age of your bishop, but to yield him all reverence, according to the will of God the Father, as I have known even holy presbyters do, not having regard to the manifest youth [of their bishop], but to his knowledge in God; inasmuch as “not the ancient are [necessarily] wise, nor do the aged understand prudence; but there is a spirit in men.” (Job 32:8-9) "
It seems he is using Timothy as an example of a young bishop, since Paul said to Timothy, "let no man despise thy youth"' and Ignatius exhorts to "obey the youthful bishop". Yet, there is no place Timothy is called a bishop, presbyter, or overseer in the scriptures. Timothy was an evangelist. The proper meaning is messenger. Evangelists were teachers and preachers, but were emmisaries for the apostles. He worked directly under Paul. Timothy was Paul's emissary or messenger. Timothy was never a Bishop in the scriptures. Evangelists simply served Apostles, at their wish.
So he sent into Macedonia two of them that ministered unto him, Timotheus and Erastus; but he himself stayed in Asia for a season.
The offices can be seen in Ephesians 4:9-11, in this list evangelist and presbyter are separate, but the word Bishop isn't listed. So it seems Ignatius confuses evangelist and Bishop. It is as simple as that really. The unstable and unlearned can make mistakes. Catholics simply confuse two offices. The evangelist was likely to be a temporary work since the apostles work would end, so their messengers would end. Of course we can have messengers under churches. They don't work directly under an Apostles authority as Timothy or Titus did.
Once Timothy ordained elders there is no evidence he continued as an overseer or he was made a Bishop. The Apostles and their companions didn't become Bishops after ordaining presbyters.
Acts 14:23
And when they had ordained them elders in every church, and had prayed with fasting, they commended them to the Lord, on whom they believed. (then they left)
There is no evidence evangelists were made Bishops.
So he sent into Macedonia two of them that ministered unto him, Timotheus and Erastus; but he himself stayed in Asia for a season.
Since Timothy and Erastus were sent to a region that came to have Bishops, whether ordained before or after this passage, they were not considered Bishops of those cities. They were not Bishop in Ephesus since they had Bishops/presbyters in Acts 20 where presbyters were called overseers, and the cities of Macedonia had Bishops, notably Philippi, it is doubtful Timothy was Bishop in the cities he previously worked.
Phil 1:1 Paul and Timotheus, the servants of Jesus Christ, to all the saints in Christ Jesus which are at Philippi, with the bishops and deacons:
Philippi is in Macedonia where Timothy was sent. It isn't likely Timothy was Bishop in Ephesus or Philippi. Catholics just made up a bad theory. It is obvious he didn't believe in the Papacy. since he says the bishop stands in the place of God. Since he wrote from Rome about all Bishops, all Bishops stand in the place of God in his view.
Acts 20 is key in many respects since it proves many points. When the presbyters/overseers were called from Ephesus, Timothy was already with Paul, he wasn't one called from Ephesus. There are other point Acts 20 makes against Catholic theory.
Papias
Papias only uses presbyters and never mentions Bishops. They were used interchangeably so he just used one is a good guess. He lines up with Clement, Polycarp.
Iraneus
Iranaeus seems to be a deceptive teacher, his writings misquote passages quite a lot. In book III he says concerning Acts 20 that presbyters and Bishops came from Ephesus and those cities adjoining. The text only lists presbyters being called, but then calls those presbyters overseers in Acts 20:28. Acts 20 doesn't differentiate. So, Iraneus lied. He also lied in saying the overseers of surrounding towns were called. There is no evidence of this. It is possible but not stated.
He is clearly embellishing Acts 20.
For when the bishops and presbyters who came from Ephesus and the other cities adjoining had assembled in Miletus, since he was himself hastening to Jerusalem
Even if he was correct that area presbyters were called, it would disprove Catholics,
Timothy wasn't a bishop in Ephesus, it is clear from this.
There would be multiple bishops, not one in a given city.
There were no arch- bishops called.
All presbyters would be made bishops (overseers) by the Holy Ghost.
All overseers would feed the church, verse 28, so all presbyters would shepherd and oversee.
Catholics need it to mean the adjoining cities to have an explanation for multiple overseers in the city. The text doesn't read such, so they embellish.
The ordination process is also affected. In scripture one man didn't ordain elders, Acts 14:23 says a plural number ordained the elders. The word elect or ordain is plural in the text.
And when they had ordained (elected) them elders in every church, and had prayed with fasting, they commended them to the Lord, on whom they believed.
Both they and ordain is plural, so one man didn't do it. In Acts 14:23 there were no bishops yet to ordain. So, a singular bishop for that really isn't necessary.