In this article I'm not making a comprehensive case for Jesus mythicism, so much as I'm talking about the debate between mythicists and historicists, at a meta level. I'm talking about how we should understand the debate, what the burden of proof is, and so forth. So that's why you'd find "holes" in the article.
Wells, Carrier, and Doherty deal with Tacitus and Josephus, including the specific points you made about them. Carrier had a back-and-forth with Tim O'Neill, that atheist blogger.
For example, it's unlikely Tacitus could have checked official records, because the capitol's libraries had burned down twice.
Carrier argues that that line in Josephus looks like an accidental interpolation. Doherty discusses both passages in Josephus at great length. Doherty points out the phrase "called Christ" is found in Matt.1:16 and John 4:25, so it was in use by Christians. Also, that line in Josephus is suspicious because the reference to Jesus comes first, before the reference to James, even though James is supposed to be the subject at issue. The reference to Jesus would be just an afterthought as a way of identifying James. Only a Christian scribe would consciously or unconsciously give pride of place to Jesus in that way.