Did Historical Jesus Really Exist?

On December 18th, the WaPo published an article challenging the historical evidence for the existence of Jesus by Raphael Lataster, a historian who claims Jesus didn't exist. It seems an interesting topic for discussion.

Lataster's main claims about the historical documents are:

The first problem we encounter when trying to discover more about the Historical Jesus is the lack of early sources. The earliest sources only reference the clearly fictional Christ of Faith. These early sources, compiled decades after the alleged events, all stem from Christian authors eager to promote Christianity – which gives us reason to question them. The authors of the Gospels fail to name themselves, describe their qualifications, or show any criticism with their foundational sources – which they also fail to identify. Filled with mythical and non-historical information, and heavily edited over time, the Gospels certainly should not convince critics to trust even the more mundane claims made therein.

The methods traditionally used to tease out rare nuggets of truth from the Gospels are dubious. The criterion of embarrassment says that if a section would be embarrassing for the author, it is more likely authentic. Unfortunately, given the diverse nature of Christianity and Judaism back then (things have not changed all that much), and the anonymity of the authors, it is impossible to determine what truly would be embarrassing or counter-intuitive, let alone if that might not serve some evangelistic purpose. 

The criterion of Aramaic context is similarly unhelpful. Jesus and his closest followers were surely not the only Aramaic-speakers in first-century Judea. The criterion of multiple independent attestation can also hardly be used properly here, given that the sources clearly are not independent. 

Paul’s Epistles, written earlier than the Gospels, give us no reason to dogmatically declare Jesus must have existed. Avoiding Jesus’ earthly events and teachings, even when the latter could have bolstered his own claims, Paul only describes his “Heavenly Jesus.” Even when discussing what appear to be the resurrection and the last supper, his only stated sources are his direct revelations from the Lord, and his indirect revelations from the Old Testament. In fact, Paul actually rules out human sources (see Galatians 1:11-12). 

Also important are the sources we don’t have. There are no existing eyewitness or contemporary accounts of Jesus. All we have are later descriptions of Jesus’ life events by non-eyewitnesses, most of whom are obviously biased. Little can be gleaned from the few non-Biblical and non-Christian sources, with only Roman scholar Josephus and historian Tacitus having any reasonable claim to be writing about Jesus within 100 years of his life. And even those sparse accounts are shrouded in controversy, with disagreements over what parts have obviously been changed by Christian scribes (the manuscripts were preserved by Christians), the fact that both these authors were born after Jesus died (they would thus have probably received this information from Christians), and the oddity that centuries go by before Christian apologists start referencing them.

He has some other things to say, and a number of links within the above text as well to other authors, but that's the gist of it. This is a new argument to me, though I've heard rumblings of it before. I've heard Christian apologists make the opposite claim that, while clearly the miracles and any supernatural parts can be disbelieved, that the case that Jesus was a historical figure is clear for all to see. I actually haven't looked that much into it, though. Any have any good sources on this? Any thoughts about it?