Jewish historian Josephus records in Antiquity of the Jews Ch. 20,
Certain of these robbers went up to the city, as if they were going to
worship God, while they had daggers under their garments; and, by thus
mingling themselves among the multitude, they slew Jonathan [the high
priest]; and as this murder was never avenged, the robbers went up
with the greatest security at the festivals after this time; and
having weapons concealed in like manner as before, and mingling
themselves among the multitude, they slew certain of their own
enemies, and were subservient to other men for money; and slew others
not only in remote parts of the city, but in the Temple itself also;
for they had the boldness to murder men there, without thinking of the
impiety of which they were guilty.
And this seems to me to have been the reason why God, out of his hatred to these men's wickedness, rejected our city; and as for
the Temple, he no longer esteemed it sufficiently pure for him to
inhabit therein, but brought the Romans upon us, and threw a fire upon
the city to purge it; and brought upon us, our wives, and children,
slavery - as desirous to make us wiser by our calamities.
The reason the Temple was destroyed may have been because it was no longer clean, no longer suitable to be inhabited by God's presence.
In addition to murders occurring in the Temple, there had been corruption in the priesthood in the decades prior. Combined with Roman rule and oversight of the whole city, with hooks in the political and religious system, the Temple may have been destroyed because of its unsuitable state for the divine presence.
Likewise, Josephus records in War that the revolutionary party of the Zealots, joined by the Idumeans, who were Edomites forcibly converted to Judaism, slaughtered 8500 in the Temple outer courts. Josephus suggests this was an irrevocable pollution of the Temple:
The Zealots also joined the the shouts raised by the Idumaeans; and
the storm itself rendered the cry more terrible; nor did the Idumaeans
spare anybody...and acted in the same manner as to those that
supplicated for their lives, as to those that fought them, insomuch
that they ran those through with their swords who desired them to
remember the kinship there was between them and begged of them to have
regard to their common Temple. There was no place for flight nor any
hope for preservation; they were driven one upon another in heaps, so
were they slain. Thus the greater part were driven together by force,
as there was now no place of retreat, and the murderers were upon
them, and having no other way, they threw themselves down headlong
into the city, undergoing a more miserable destruction, in my opinion,
than that which they avoided, because it was voluntary. And now the
outer Temple was all of it overflowed with blood; and that day, as it
dawned, saw eight thousand five hundred dead there.