You wrote: "In short, because Josephus wrote so little about Jesus, Jesus's followers and witnesses were wrong about their testimony?"
In short, no. Josephus wasn't a Christian. He wasn't interested in Jesus. He was a Priest of the Temple and a Jewish general writing about Judaism for a Roman audience.
You wrote: "Any chance he didn't write much because others who were actual eyewitnesses already were on record? Any chance he wrote about those things which had not been recorded by others and which he had better direct knowledge?"
Not really, no. He gives us the speeches of Noah and Moses and Aaron and an endless array of other stuff that he didn't have any knowledge other than traditions. He wasn't a Christian and wasn't interested in Christianity. He wasn't hostile to it either. He is sympathetic to Jesus as a wise man, and in his writing about the execution of James, brother of Jesus, he clearly thinks that the execution was a travesty of justice. But he doesn't comment on the merits of Christianity, not because Christians were out there writing their Gospels, but because he was writing about his own Jewish culture and beliefs to a Roman audience. A couple of Christian figures pop into the narrative, but Christian theology is nowhere on the radar screen.
You wrote: "It seems in order to conclude that Jesus wasn't Jesus, you need to assume an awful lot about a writers motivations." No. One need only look at the text in the context of the rest of the thousands of pages of Josephus, and it is clear that Josephus was not a Christian. If he thought Jesus was the Messiah and raised from the dead on the third day, he would have been a Christian. He wouldn't have passed over that in one single sentence and then gone on for another 500 pages about Jewish faith and practice.
Josephus tells us that Jesus was Jesus, a real man renown for marvels and with a persistent following. The bones of the Christian history are there, and Josephus isn't hostile to them. But it's just not possible that a man writing a religious history, and writing about religion, avers the Messiah and the Resurrection in one single sentence, and then ignores it throughout the entire rest of his 2000 page work. Jews for Jesus don't do that. Nobody who avers that a man is THE MESSIAH, who is writing about Jewish religion, mentions The Messiah once, in one sentence, and then never mentions it again.
That's not the way anybody's religious mind works.