You know, let me defend the "incompetence" of the exorcist a little, not trying to justify, only explain.
It is pretty clear that the current model of how Tsuzura temple operates is a "peace model", they supposedly had the unrestricted backing of the government, and they are the representatives of the gods' of the land, there is no war and the closest thing to a "true" enemy faction is too unorganized and descentralized.
Only their highest ranking members are put on execution missions against human criminals ( I would bet this is by design of Oriobana, who is a teacher at heart ), so chances are many students don't even know there are human criminals that use tsukumogamis.
When I say "unrestricted backing" i want to emphasize the "supposedly", there is clearly conflicting interests between the government and the temple, chances are there is conflicting interests inside of the government, but Oriobana as the leader doesnt have experience with reaches the stupidity, greed, arrogance, disregard and ignorance of the government side has.
I wouldn't put it past that the existence of the shards was something that was "hard fought" to know by the government, but what they see isn't the absolutely dangerous things they are... but a chance at godhood.
You just need to give a glance at Sekimukai's narcissism to see what he thinks of himself; he is a leech that uses others without regard to their safety and claims absolute ownership over anything they accomplish, is clearly ready to discard them, and is a creep.
Point is... Tsuzura should have been more prepared, and they did have a few things set up, but the world they live in now simply didn't gave them the experience to such an attack (which mind you, i think is a good thing, war isn't a good thing)... a attack mind you that was planned from the inside, Sekimukai has info that the curse users wouldnt be able to get, that is why this attack worked so well.
Oriobana in particular... I think she is more than anything a teacher, not a perfect one, I think she makes mistakes on that, but she clearly directed Tsuzura to be a safe haven to children with power and tsukumogamis, unfortunately she didnt have the foresight to predict this... just like it happened with Madarai, and she is paying the price for it.
Now that I write all of this... I see that Hamada-sensei had all of this planned, from Oriobana's personality and the early takeover of the temple, to the "relaxed" way the temple is run.