While I do understand the basis of the blockchain, I did not read the BTC whitepaper until 5 minutes ago.
If we do compare the standard distributed database with the private blockchain, there are IMO completely something different. Almost all DDs (distributed datase) are based on replication or duplication, which both have their pros and cons. If in a duplication the master has been compromised, the slaves will also be compromised. If in the replication a hacker intrudes into one of the database, the fault record would/could be copied into the others.
In a private blockchain, the consensus mechanism is still in place (even in blockchain without a token). A new block is only approved if their is consensus in the complete blockchain. In the original whitepaper the first transaction in the new block, is the transaction which rewards the coin to the creator of the block, in a token less blockchain, this would not be necessary.
But to respond to your question:
Can you clarify exactly what you mean when you say “blockchain”?
A network of nodes, where information (important to the use case) is gathered and stored in block. Each block will have a hash based on the content of the block. The next block in the chain will refer to the previous block, creating the chain. New blocks are proposed to the network and will only be accepted if there is a consensus between the nodes in the network. When one of the older blocks is tampered with, it will not be accepted by the network cause the chain will be broken ( the hash will be changed). To succesfully attack the blockchain, the hackers must be capable of hacking enough nodes, and rewrite the complete block after the tampered block on all nodes, before one of the other nodes creates a new blocks. You can add to this the timing mechanism, the smart contracts, and so. By doing this there is no need for a third party to trust each other but the trust is based in the cryptography of the blockchain.
This is short what I would call a blockchain.
In my opinion each of this could be perfectly feasible with or without a native crypto currency. And not all of the above could be done on a distributed database.
If I do have some things wrong, please do not hesitate to correct me. I can only learn from it.
The mechanisms you mention regarding consensus are what distinguish a private database from a blockchain because a private database obtains consensus via controlling the nodes where a blockchain is uncontrolled and wide open to attack. It's that open nature, to me, that separates a blockchain from a database. The only thing protecting it is the incentive provided to the block producers to be rational actors in the best interests of themselves and the network. That incentive only comes through the distribution of a valuable token. I have yet to see any other mechanism for providing that incentive. I have seen blockchains with tokens of little value be 51% attacked. If the token is not valued enough, not enough miners will mine the blockchain to reach useful, secure consensus which means it will fail. I say "mining" here but similar economic incentives exist for DPOS as well.