And I think you're right -- adding Nakamoto consensus to private blockchains makes no sense, because by definition they don't have arbitrary validators. And without Nakamoto consensus, "blockchain" is just rebranding of old and boring tech.
A distributed merkle-chain database is still pretty innovative. A good example is git. I think what many private groups want is a "binary git for transaction data". Everybody can review their own copy of the shared chain with signing of new links in the chain.
Now that isn't really a full blockchain, but is a different kind of thing from a central RDMS.
The point being that all the hype about "blockchain everywhere" came about because of bitcoin. If people are now starting to refer to just the "hash of blocks with previous blocks" (i.e. the "Timestamp Server" section of the original bitcoin whitepaper) as blockchain withOUT the proof of work/stake part, well then you really are just rebranding old tech as blockchain, because it was the proof of work part that was really the new thing in bitcoin.
Proof of work is required for distributed consensus; without that you need a centralized repository like GitHub or Linus to bless the current state of the database?
With Git for code you don't really need consensus - I can have my fork and you can have yours.