An offline trusted third party doesn't disqualify a blockchain — what role can it still play?

An always-online TTP replaces the blockchain (it can just run the database); an offline TTP can instead act as a certificate authority that fixes a known set of writers, enabling a permissioned blockchain.

The TTP's availability changes its role entirely:

• Always online → it can receive writes and verify every state transition itself. That's a trusted operator running a database — no blockchain needed.
• Usually offline → it can't verify each transaction, but it can still issue credentials up front, i.e. act as a certificate authority that decides who is allowed to be a writer. That gives you a known writer set — the precondition for a permissioned blockchain.

So "we have a trusted party" doesn't automatically kill the blockchain case; it depends on whether that party can be online for every transaction.