The bible have contains two almost exact copies of the same text, one in 2 Kings 25 and one in Jeremiah 52. One of the differences is that Jeremiah says that commander Nebuzaradan arrived on the 10th day of the fifth month, and 2 Kings say he arrived on the 7th day.
Neither gives an exact date for the destruction of the temple, they just say that he " He set fire to the temple of the Lord, the royal palace and all the houses of Jerusalem." The next date mentioned (only in 2 Kings) is "The seventh month", so presumably the destruction of the temple happened before that.
So for all we know the destruction of the temple could have happened any time between the 7th of Av and up to two months later.
However, Josephus refers to the date of Jeremiah, and says that the first temple was destroyed the 10th. And he also says that the second temple was destroyed the tenth. (Not the 9th as later tradition has it).
In the quote you show he does reflect on this coincidence, and clearly wonders if it is true. So, what is he actually sceptical about?
Does he doubt the claim that the first temple was destroyed the 10th? This is possible. He might after all be aware that another text claims the 7th.
Does he doubt the claim that the second temple was destroyed the 10th? Wasn't he an eye-witness? Well, yes, he was, and he was Titus translator, and should have remained by his side. And this is what he says about the destruction:
So Titus retired into the tower of Antonia, and resolved to storm the
temple the next day, early in the morning, with his whole army, and to
encamp round about the holy house. But as for that house, God had, for
certain, long ago doomed it to the fire; and now that fatal day was
come, according to the revolution of ages; it was the tenth day of the
month Lous, [Ab,] upon which it was formerly burnt by the king of
Babylon; although these flames took their rise from the Jews
themselves, and were occasioned by them
What he is saying is that the Romans did not destroy the temple. The Jews themselves did. Your quote above seems to say that the date was not a coincidence, and that maybe they decided to burn it down partly because it was the same day as the previous destruction.
Josephus is painting a very positive portrait of Titus, claiming that he offers to save the temple and let the Jews continue to worship there as before. If this is true of just Josephus doing what he is paid to do as propaganda I don't know. But your quote seems to me to be a part of that: Josephus making it completely clear that the Romans in no way is to blame for the destruction of the temple.