Personally, again, I have no problem with it as it is written, but who knows. Wiki can be a great first peek at a topic but remember it is a crowd sourced encyclopedia. The accuracy of the info is inversely proportionate to the controversial nature of the topic.
I noticed the snip you posted from wiki says
:
'In the Antiquities of the Jews (Book 20, Chapter 9) first-century historian Josephus states that Jesus ben Damneus was made high priest after the previous high priest, Ananus son of Ananus, was removed from his position for executing James the brother of Jesus of Nazareth (James the Just).[2] This occurred after a large number of Jews complained and petitioned the king. Jesus ben Damneus himself was deposed less than a year later.
See how when identifying Jesus of the Gospels even this contributor identified him in the more neutral way, "Jesus of Nazareth" when making his proposal. If a secular Jew writing 50 years after Jesus (as is usually presumed) were to first introduce a new character to his readers, surely he would have identified him by his surname (ben Joseph) or his home (of Nazareth). If he meant to say this Jesus of Nazareth was a would-be Messiah he would also have defined his movement as he does with a half dozen other would-be Messiahs in his works.
The wiki contributor also summarily assumes Jesus of Nazareth is the same as the brother of James the Just. That is circular reasoning, an assertion, not evidence or even an argument. The very controversy in discussion is if 'James the Just' is to be identified as the James who was killed in Josephus OR with the James who led the Christain church in Jerusalem, OR a 'brother' of Jesus of Nazareth OR none of these Jameses.
Ask yourself why would the Jews be in uproar about the death of a Christian or subsequently blame the destruction of 70 on his murder as other writers assert? Recall it was they who are described as the ones who had Jesus killed and according to Paul were actively persecuting them at this time. Even 'James the Less" (aka apostle James ben Alphaeus) was said to have been killed as a result of persecution a around this time. Let it all sink in.
It is far easier to believe Joes was making a brief interrupting phrase to identify James by his association with the High Priesthood, (the topic in discussion) He summarily describes this Jesus as the anointed High Priest who gets the position as a response to the murder of his brother. It fits Josephus' pattern and context as well as sound logic.