Josephus, the (in)famous Judeo-Roman historian, in his pivotal account of the Jewish-Roman war, seeks to persistently and systematically distance the Romans from any moral responsibility in the Temple's eventual downfall.1 This is commonly explained (away?) by portraying him as a Roman pawn, subservient to his gentile captors.2
Whilst not entirely devoid of merit, there are at least a few drawbacks to such a simplistic perspective, the most pertinent being that the Romans themselves were not exactly known for being particularly shy, let alone ashamed, of their military triumphs and victorious conquests; if anything, quite the opposite.
To make matters “worse”, there are two closely related Jewish pseudepigrapha,3 both predating Bar Kochba's revolt, one of which goes even further than Josephus ever did, by denying not only Rome's moral, but also factual involvement in the Temple's destruction, by explicitly stating the latter to have been the direct product of divine intervention (2 Baruch 6–8).4
While no acts of God are mentioned by Josephus as a possible cause for the edifice's demolition, it is nevertheless somewhat awkward to even suggest that a pious Jewish composition,5 regardless of its canonical status (or rather, lack thereof), is somehow supposedly aimed at either favoring pagan Romans, or otherwise excusing their nefarious actions and subsequent wrongdoings, towards one of Judaism's most sacred sites.
A clear trend and consistent tendency seems to definitely emerge and become noticeable; which ultimately brings us to the nagging question, contained in the post's title, whose solution has been eluding me for quite some time now, namely:
Whence this perplexing insistence on exculpating the invaders ? It just seems off.6
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] making it completely clear that the Romans [are] in no way [...] to [be] blame[d] for the destruction of the temple.
For they set the north-west cloister...on fire...and thereby made a beginning in burning the sanctuary.
They lay still while the temple was first set on fire, and deemed this spreading of the fire to be for their own advantage.
They had begun with their own hands to burn down that temple which we have preserved hitherto.
They above is always the Jews. There's little doubt whom he blames, at least from a moral standpoint. He does mention the soldier who sets fire to a window, but also says that fires were fought before that, and doesn't say that the soldier is the direct cause.
He is so explicit about whose fault it is, at least morally, that he doesn't really need to be subtle about it. He might just think it's too much of a coincidence [that both temples were destroyed on the same date].
3 Namely Second Esdras, penned around the turn of the first century, as can be glimpsed from the first verse of its third chapter, and Second Baruch, the former being part of the King James apocrypha, having been historically included in various editions of the Latin Vulgate.
4 See 2 Baruch 7:1:
And after these things I heard that angel saying unto those angels who held the lamps:
Destroy, therefore, and overthrow its wall to its foundations, lest the enemy should boast and say: We have overthrown the wall of Zion, and we have burnt the place of the mighty God.
5 I feel compelled to stress or underline here that, precisely because of its inherently or intrinsically religious nature, this ancient witness is oftentimes disregarded by professional historians, and understandably so; furthermore, due to its lack of canonicity (indeed, it is not even part of the Apocrypha), it is often overlooked by religious scholars as well.
6 For lack of a better comparison, it almost reminds me of post-9/11 conspiracy theories, suggesting the tragic event to have been an inside job, rather than facing up to the fact that, for the first time since Pearl Harbor, the US has been the unsuspecting victim of a foreign attack on its own sovereign soil (as the Ukraine currently is, but I digress). — Perhaps this would be better suited for Psychology.SE, instead ?