Well, It would be unobjective in the extreme to say that the narrative of history here is without its weaknesses. There is certainly room for examination, and that is exactly what Bart Erhman did in the early part of his career.
His belief, not without merit, is that many historians either offer deference to theological premises out of fear of offense or out of their own religious biases. His case then is aimed more at the reality that the genesis of Christianity was not, at the time, widely recognized as the important event that we have come to understand it to be now. That the grandiose things we see in hindsight were not all the apparent gems in the rough ... that Jerusalem was not Rome ... and not by a long shot.
He has nevertheless had to correct, as stated earlier with, "Was Jesus Real?", had to make the larger narrative support for Christ into a compelling and cogent explanation - even as he cautions against the tendency to read more into things that is tenable. As an agnostic, favoring neither atheism nor Christianity, he is perhaps best placed to be objective - cuffing Christian who see a manger as a palace that its actually a manger and cuffing atheists who see the genesis of a major world religion as nothing at all.
That is a roundabout way of saying there are things to criticize here. I will say that humans have a tendency to dress things up. Yet dressing up a piece of crap hardly hides that there is a piece of crap underneath (Oh yeah, Al Qaeda really WAS in Iraq hiding the WMD!!!), and that particularly over time people are adept at unwrapping the dressing to see crap clearly for what it is.
Paul was human. He has his biases as we all do.
Humans are humans.
And just as extreme atheists become consumed by their biases to become Mythers, there are Christians who tend to view these historical notes with a same fanaticism in which NOTHING wrong can be found. The inerrent view of religion is, IMHO, just as bad as the only errant view of religion.
Indeed, as one grows in the faith, most ministers are quick to point out that no an, not even Prophets and Apostles, are perfect. That as we study scripture, we will come to see the men of the cloth as the humans that they are. Each of the Synoptic gospels was written for a specific audience for a specific reason.
The one real difference between this narrative is that its ultimate truth isn't the words, its the relationship it inspires - where the truth of the text comes out in unexpected ways.