The thing about Josephus is that he was not a Christian, but a Jew and therefore would not report about Jesus because of his faith in him. Also, he would have been in contract with first century Christians as well as other people that may have witnessed some of it soon after the fact.
Sadly, he was not convinced of who Jesus was, and died as lost as anyone. But, he left historical writings of the time that gives a testimony that Jesus actually existed.

