How did the British rulers in India fail to check the treasures hidden in temple vaults? Was this mistake their first mistake or is there more to it?

The Angrej were vandals of such reknown that no few contemporary (non Angrej) travellers remark upon their acts.
The French explorer Louis Rousselet savaged British ‘historians’ in an extensive travelogue during the 1860s: “The English are busily employed reducing the need for archaeology… of Indian history. Already all buildings… are reduced to rubble and the same fate is reserved for the rest, even the Jain statues. When I returned in December 1867, the trees had been cut down, the statues shattered by workmen’s picks, and the ravine filled with rubble of the palaces of the Tomars, Chandelas, idols of the Buddhists and Jains.”
Sir Alexander Cunningham, the father of modern Indian Archaeology, mentions countless such examples of looting and vandalism in his texts. Maurya pillars were lost when Angrej officials used them for ballast & construction. Ancient temples were torn down for loot.
Except for temples either directly under princely control or far too important for the community to be stripped completely, the Angrej and their Firang allies carried out looting on an incredible scale. The Hope diamond used to be in a Telugu temple. And let's not start on the Missionaries.

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