Actually I do not see that Blogetery necessarily violated the TOS.
Running a blog website itself does not violate the TOS. Someone used one of the blogs for illegal purposes, but this does not necesarilly mean a breach of contract or TOS on blogetery's side.
Imagine this: You rent an appartment. Someone breaks in your apartment and does somthing illegal in it - selling drugs. When you come home your landlord has switched locks and all your property is gone. The landlord says by selling drugs you violated the contract and he has now cancelled the contract without further notice and removed your belongings.
Is he within his rights? Certainly not. Because someone else doing something illegal with something your rented does not constitute a violotion of the contract by you.
Or you could argue like this:
BurstNet has allowed Blogetry to run the website for several month. BurstNet knew that Blogetry was running blogs. BurstNet also knows that it is possible that third parties post illegal content on blogs and this is impossible to prevent. Therfore BurstNet can't suddenly claim a breach of contract when something happens that BurstNet knew all along would very likely happen. Maybe even knew when they signed the contract.