There seems to be an assumption that “the Church” is equivalent to a particular institution, and thus “the early church” must be a singular institution to which all of “the fathers” belonged. Since Ignatius talks about “the bishop,” which he does, then the institution must be one of a sort of apostolic succession, or so the saying goes.
There are lots of problems with this. Ignatius’ bishop is a local eucharistic president (see Lowrie and Sohm on this) and not a jure divino bureaucratic institution. “The early church” consisted of several institutional churches, mainly divided along national lines, which then became theological lines: Antioch vs. Alexandria, Rome, Byzantium, Persia, Copts, etc. “The fathers” is a variegated collection, limited to those authors (by no means a basic representation of pastors) whose works we still have. Many of the church historians (Eusebius, Socrates, Sozomen, Theodoret) belonged to groups considered heretical or partially heretical by today’s “Catholic” Christian. Origen has become a heretic, but was loved by most of the fathers (Ambrose, Nyssa, Augustine). Cyril of Alexandria probably should be held with more suspicion, and Nestorius should probably be held with less. In fact, the Alexandrian bishops all tended to be bad dudes.
And of course, what we usually call “the fathers” tends to be limited to 3rd-6th centuries, with the Second Council of Nicaea coming in quite late in 787. The truly “early church” are the Jewish Christians (Petrine and Jacobean Christianity) who are almost wholly lost now; there are a few exceptions to be found among the apostolic fathers and the Syriacs.
The Reformers dealt with this problem of “the Church” by appealing to the visible/invisible distinction. This didn’t mean elect and reprobate, as it later came to mean, but rather the Holy Spirit and polity, with the goal that the two would work together, but the understanding that they did not always do so. Thus, “the early church” and “the Church” are two very different things.
In fact, sometimes “the Church” can be best seen through actions of “the State” restricting those of the clergy. I would put Theodosius II in this category when he arrested both parties at the Council of Ephesus and forced them into concord. Charlemagne is also an example of this, when he rejects Nicaea II and calls the Council of Frankfurt. This was also how the Papal Schism was settled at the imperial-powered Council of Constance.
We are still moderns though, not ready for the medieval Christendom of kings (the true Christendom), and thus we turn to hierocracy, since it still promises to subvert the state through the church (which again is not actually “the Church” at all).