There are some interesting points here. That Julian would point to the lack of mentions from non-Christian writers from the times of Tiberius and Claudius (the lifetimes of canonical Jesus and his first apostles) may reflect the same concern about the received TF (even after cleaning) and Tacitus that many moderns have.
That is, any mention of Jesus or his first apostles after their lifetimes would be contemporary with or later than the Christian gospels. It would be as impossible in the 4th Century as it is today to eliminate dependence of any such "late" mentions on Christian sources.
There's nothing in Julian which survives that makes him a 'mythicist,' although enough is lost and what survives may be subject to interpretation to leave open the question of how confident he was that Jesus was a real man who actually lived.
From a rhetorical point of view, a "nobody" Jesus who died a shameful criminal's death and whose remains were lost might well be more useful to disparage Christian beliefs than a Jesus who never lived at all. As Celsus's Jewish character put it, he could rebut Christians using only their own writings; nothing more was needed.
Conversely, any personal doubts Julian might have had about the historical situation would not rise to the level of an argument unless he could prove some rational foundation for such thorough doubt. That would be difficult in the absence of mentions of the Christian story, including adverse mentions, contemporary with the first apostles.
Although I have no competence to enter a tactile conversation about the Slavonic Josephus, I think Maryhelena raises a good point about the possibility of a TF that doesn't mention Jesus. Received Tacitus doesn't; maybe Josephus didn't either.
As some readers may know already, my own view of both Josephus and Tacitus is that they are offering an etymological note about why Christians call themselves that. As Steve Mason sometimes points out, people may well have wondered why anybody would be called "the oily one(s)." Reason enough, then, for a writer in the know to tell his readers the derivation.
For that purpose, it is relevant that a man whom the Christians admire supposedly used the handle
Christ. It is irrelevant what that man's real name was, and received Tacitus omits that factoid.
Maybe Josephus did, too. If we look at a fairly typical (but etymologically oriented) estimate of a hypothetical genuine TF, the omission of the name
Jesus makes little difference.
Jesus There lived about this time, a wise man. He drew to himself many Jews, and also many Greeks.
He was the ‘Christ.’ When Pilate, prompted by our leading men, condemned him to the cross, those who loved him from the beginning did not forsake him, For he was seen by them alive again on the third day. The tribe of the ‘Christians,’ so named after this man, survive to the present day.