Ah, my error in having primarily the Church Fathers in mind, since it's from them that I draw most of my proof for the existence of the Catholic church because they were the most interested in writing about it.
Indeed some the above wrote about Christianity but why would they be interested in giving a detailed account about it? I.e. Josephus who was writing about the history of the Jews, Seneca, a Stoic philosopher...
While we are opposed in our alignment, surely you must understand the danger of accepting a condition where any entity is allowed to define itself - As SnakeDoc rightly declared, it "creates something of a feedback loop."
In my travels through history, much like yourself, I could find no real evidence of a structured church until late 200AD - In fact, the modern writers you quoted were quoting primarily 300AD+ sources...
Not that ALL Roman Catholic sources are impeachable - Eusebius (I think) quoted Thallus, another quoted Phragon, both first century sources... But these (Thallus, Phragon) bring no evidence of priests and robes, popes and crosiers to the game.
(Bear in mind that I am a fan of Eusebius, primarily because of his known access to the Alexandrian Library)
I am more likely to take the offhand remarks of neutral parties, or even critics, as evidence before I would take the machinations of the party in question itself - Especially when that entity has the power to rewrite history - which is a proper position to take.
As an example, If one sees history outside of what Rome may have been able to manipulate, James the Just (Jesus' brother) is the notable successor to Jesus.
As another example, an early sect which the Roman church despised (as they are critical, I pay attention) - The Nazarenes - were visible into the fourth century, when the Roman Church assumed the name. These folks, for all that the Roman church hated them, followed Christ's path more closely, embracing God's Holy Days and Hebrew (not Judaism) traditions, while accepting Christ's message. A completely different path than the Roman church subscribed to, and by then, demanded.
The mystery of iniquity walked the halls of the Church even in Paul's day - He saw it happening then. Therefore, no historic church has the convenience of resting on it's holy laurels - Proof of claim is even made more necessary - And that proof is not there.