There are two things you have to trust, which I mention in the post.
1. That the node is not denying writes. (What's sometimes in blockchains called censorship.) If the node just didn't run your request, you'd be locked out with no evidence. It's not unsolveable, though - you could combat this by using a 3rd party proxy which records the traffic, and thus can back up your claim that you're being censored.
2. That the node won't just shut down. To combat this, you'd need a procedure for moving to a new hosting node. Not impossible either, but you'd need to decide for your userbase what kind of process would make them most comfortable. The blockchain state is there to be cloned at any time.
At this point the validators compare notes and gossip a proof that the node's been acting up. And then they need to elect another node, which requires... consensus!
Oh and the central node can also become unavailable and DDOSed, how are you going to add transactions to the ledger then?
If you have hundreds of known, diverse nodes distributed globally (run by very different institutions like Harvard, Bank of America, the Vatican, the Red Cross, the Gates Foundation, the House of Saud, etc. etc.) they can all watch each other, and the odds of them all going offline at the same time or cooperating to deny a specific user's transactions are very low.
This is essentially how CT-logs work, where we consider it sufficient to detect a falsely-issued certificate, even though we cannot prevent such issuance.
