Well, I do think that that is exactly things that are important.
You'd want uncensorable, to give transparency to the patient. E.g. a hospital can't add a record without you knowing, that you might not want an insurance company to have access to later.
Immutable, as a patient you would want to know exactly what your data looks like at any given time. Again insurance is a good example.
If the blockchain is private, could it not be part of the implementation that does the access checking? Can't part of the ledger be unencrypted while other parts are not?
It might be wishful thinking. It's just an idea I have floating in my head, as an actually useful real world implementation for a blockchain.
I shared the link in another comment, but you might find it interesting as well:
https://journals.sagepub.com/doi/full/10.1177/14604582198663...
