And when you tell the judge that you don't even have the ability to take down the offending material, what then?

You can absolutely take down the offending material; you'll just lose the collateral (you can also transfer the contract to someone else but that will likely not be legal in all jurisdictions). However, as the contracts are made between the clients and miners directly, miners can choose to charge anonymous/untrusted clients more to make up for this potential eventuality.

But you do, as far as your own hosting of that data is concerned - You shut off your node and delete your data store.

out of curiosity, who is this judge talking to again? A storage node?