The fundamental rule is that the government is entitled to evidence.
If a church, for some insane reason, kept written records about what was said in the confessional, and a court order was somehow granted to order the church to turn over their records (ignoring privilege), they would need to turn those over.
If someone destroyed them, the DOJ would not need to shut down the church. They would charge the person who destroyed them with the appropriate crime, and then everyone would move on.
If they refused to turn them over, but the DOJ believed they were still on premises, eventually the DOJ would get the local sheriff to physically enter the building and take the records. There would be no need to shut down the church.
In that situation, ideally the records are all in individual files, and the constable only needs to take one. If the head priest said "nyah nyah, we mixed them all up, ha ha!", then all would be seized, and someone would need to go through and find the relevant information.
By keeping all the secrets of their "clients" written down, the church has put them all at risk. If one priest knew the secret only in his head, the government could compel him to testify, and that still wouldn't require shutting down the church.
The nature of computers is that they are simultaneously typewriter, gold bars, and file cabinet -- they are needed for work, they are valuable, they contain lots of information. When the government needs to seize evidence, they cannot slice out the "file cabinet" component from the other components. The fundamental rule is that the government is entitled to evidence.
EDIT, and maybe this is what I should have said instead of all the stuff above: Lavabit's only defense against seizure of its keys if a court order demanded them was to refuse to provide them. I.e., break the law. I don't know how else that can be read except as a promise to break the law on behalf of their clients.
Second edit: And fundamentally an unkeepable promise. I could, in theory, promise to not repeat the secret you told me and that I hold only in my brain, no matter how long I sit in jail. I cannot promise that the DOJ cannot seize my written record of that conversation, because they can bust down the door to get it.