peacefulpete, but in the passage you refer to Joseph does not say Jesus was the anointed or "was the Christ" or "was Christ", but instead Joseph says "... Jesus, who was called Christ ..." I and a number of scholar believe he says that way because he (a non-Christian Jew) did not believe that Jesus (the one that Christians believed was the son of God) was the foretold Messiah/Christ, but only someone whom some people called the Christ. If Josephus was speaking of the Jewish high priest (someone who obviously received an official anointing) in that passage then he would have no difficulty in directly calling him an anointed one, since all of the Jewish high priests received an anointing as a priest.
You are the only person whose words I have read of saying that passage of about " ... Jesus, who was called Christ ..." was about the Jewish high priest instead. None of the biblical scholars and historical scholars whose words about that passage I have read attribute it to being about the one you say it is about (or even say it might possibly about such), likewise for other highly credible sources, including the Encyclopedia Britannica. [If Earl J. Doherty also gave the explanation you gave, then I was not aware of it.] Likewise when I listened to NT scholars and historical scholars in documentaries on TV talking about that passage I did not hear any of them give the explanation you gave.
Can you provide a scholarly source which states your interpretation of that passage as not being about the Jesus whom Christians believe is Christ/Messiah and the son of God? But perhaps you are first person to come up with the idea and perhaps you correct, but thus far to me that idea is incorrect, even though I think you are right that Josephus does mention the high priest a few lines below. I used to have a hardcover book copy of a popular English translation of the writing of Josephus but after reading parts of it and photocopying parts of it, I resold the book.
Update: After I made this post (prior to making this update) I noticed that you made a post which quotes several lines of what Josephus wrote, including "... and made Jesus, the son of Damneus, high priest ...." But I now notice an additional problem with your reasoning on this matter.
Notice that your quote of Josephus says "... Agrippa took the high priesthood from him, when he had ruled but three months, and made Jesus, the son of Damneus, high priest." Notice that in those words it indicates that Jesus, the son of Damneus became high priest (and thus became annointed) at least three months after the events about "... the brother of Jesus, (who was named the (annointed) Christ,)
whose name was James, and some others, [or, some of his companions]; and
when he had formed an accusation against them as breakers of the law ..." As a result the Jesus, the son of Damneus (who became high priest) was a different Jesus than "... the brother of Jesus, (who was named the (annointed) Christ,)
whose name was James"! I have read and heard biblical scholars say that the Jesus who was a high priest (including the one you mention) was not the one referred to as "... the brother of Jesus, who was called Christ, whose name was James ...".