I have not read Alice Whealy’s discussion of the Testamonium Flavianum in its entirety.  In fact, I have only read the abstract, the introduction, and a small portion of the text.  However, I have read enough that I think I can give a reasonably useful answer to your question.  I get a sense from your question that you have not read what Whealey wrote, but what a critic of the reliability of the Bible has said about her article.  This is unfortunate, as I detect bias.

Here, in a nutshell, is what Whealy concludes.  First, she reminds readers of the work of the earlier scholar, Shlomo Pines. In 1971, Pines published a groundbreaking study of a 10^th century Arab fragment of Josephus’ Antiquities and did a comparison of the Greek textus receptus of Josephus to the Arabic fragment.  Pines concluded that the Arabic fragment of Josephus was probably closer, if not identical, to the original of Josephus. In other words, he proposed that there was indeed a Christian interpolation [material unfortunately added to the original work of Josephus by a Christian believer trying to boost its “proof” of Jesus] of Josephus. Shlomo noted that the Arabic version of Josephus was missing parts of the Greek textus receptus which was “fishy,” because it seemed more like a Christian commentary than something Josephus, a non-Christian, would have written.  He proposed that the shorter Arabic version is closer to what Josephus wrote. It is fair to say that many have accepted the work of Pines as likely to be correct.  If we take Pines at his word, Josephus did indeed mention Jesus of Nazareth, including that he had many followers and that he was crucified, but part of the Greek text is an interpolation. Note: At the bottom of this article, I have the relevant section of Josephus Antiquities 18:3,3, with the proposed original of Shlomo, and with the proposed interpolated material in italics.

Whealey attempts to update and to improve on the work of Pines. She notes that we actually have two different non-Greek manuscripts of Josephus’ Jewish Wars, including the Arabic manuscript from the 10^th century, which Shlomo analyzed, but also a Syriac version from the 12th century.  Whealey argues that the Syriac manuscript, which is closer to the Greek textus receptus of Josephus, is most likely also closer to the original of Josephus. She does not deny the role of Eusebius, but she does conclude that Josephus did in fact mention Jesus of Nazareth. What is new from Whealey is that she believes the Syriac is closest of the three ancient manuscripts to what Josephus originally wrote.

Like I have said, I have not read this work, so it is hard for me to decide if Whealey argues compellingly.  However, let me speculate that if I were to read this treatment, I believe that it is likely that her argument would be suggestive, if not compelling.  Here is the important thing for us to recognize about the work of Whealey.   We should note what Whealey does not do.  Her article is NOT a repudiation of the Testamonium Flavium.  In fact, Whealey argues that the current Greek text is probably closer to the original than Pines has suggested.  Whealey’s work is NOT a repudiation of the commonly accepted understanding that Josephus mentioned the ministry of Jesus or the means of death of Jesus.  If anything, it is a stronger support, both for the reliability of the Bible, and for the original textus receptus  of Josephus than the work of Pines in the 1970s implied.  She did not, as you inquire, refute the idea that the Testimonium Flavium supports the authenticity of the New Testament.  In fact, she does quite the opposite.  She makes the case for the reliability of the New Testament stronger, not weaker.

I am a busy person, so please accept my apology that I have not read the entire work of Whealey, but what I can say is that anyone who claims her work undermines the belief that Josephus mentioned the ministry and means of death of Jesus is not speaking truthfully.

John Oakes