Were the earliest Christians united under a papacy?

In a recent video discussing his conversion to Roman Catholicism, Joe Schmid said:

"And then, totally unsurprisingly, given their defective courtless system for church governance, Protestantism turns into a schism fest....You know, Christ wanted people to remain united while preserving the whole truth full and inviolate. Any sensible guy, much less the omniscient Creator of the universe, could foresee that the Protestant system was not going to work for that....Commonsensically, if Christ intended to establish a unified worldwide church that would preserve one faith across nations and centuries, a continuing teaching office is exactly the sort of institution we should expect him to provide. And then when we turn to history, we discover that this is exactly what the earliest Christians adopted."

In the second century, Celsus wrote:

"Christians at first were few in number, and held the same opinions; but when they grew to be a great multitude, they were divided and separated, each wishing to have his own individual party: for this was their object from the beginning....being thus separated through their numbers, they confute one another, still having, so to speak, one name in common, if indeed they still retain it. And this is the only thing which they are yet ashamed to abandon, while other matters are determined in different ways by the various sects." (cited in Origen's Against Celsus 3:10, 3:12)

And see here regarding Origen's response to Celsus, which further illustrates the absence of a papacy in early church history. Or see here regarding the anti-papal ecclesiology of Ignatius of Antioch, among many other examples that could be cited.