I continue to maintain that Josephus and Acts are sufficient grounds for the presumption of an historical Jesus.
I have serious doubts about Acts. There's no extant reference to it until the latter 2nd Century, and there's much about it to suggest that it was written as a polemic - possibly in response to Marcionism.
Amazon.com: Marcion and Luke-Acts: A Defining Struggle (9781570036507): Joseph B. Tyson: Books
As for Josephus, I agree that there is some evidence of a person named Jesus, but it doesn't support the extrapolation that the Gospels accounts are about that person. In terms of the historical record, everything points to James being more important than his brother.
Do you agree at the least that his adulatory comments about Jesus's deity in the
Testamonium are probably not genuine?
What that leaves are two references in Josephus.
(1) A reference to the stoning of James, who the author considered important enough to mention, and who had a brother named "Jesus" that includes a debatably authentic reference to this brother being called "the Christ". The only historical detail gleaned is that there actually was a James, who had a brother named Jesus, and that James and his followers were stoned by an unlawful session of the Sannhedrin. The same passage references another person named "Jesus" as well, who replaced Ananus as high priest.
(2) The problem of piecing together something from the probably altered
testamonium using manuscripts that are no older than the 10th century and extant references no older than the early 4th century.
My own opinion is that Josephus was reporting on the religion that had emerged by the time he was writing
Antiquities and was simply reporting the beliefs of the group of early Christians that had emerged by that time, which was later altered to make it sound like he was endorsing their beliefs. If he had been speaking of the same "Jesus" that he mentions as the brother of James in the other passage, it seems awfully conspicuous - given Josephus's thoroughness - that he doesn't mention that this "Jesus" is the same guy who was the brother of the dude whose illegal trial and stoning ended the tenure of Ananus as high priest.
My two cents.