Note that storing health records is a very dumb try of using blockchain. No wonder that you have trouble seeing how blockchain would be of any help, though probably for reasons different than the proper ones.

Blockchain is a service that adds a (cryptographic) timestamp to documents, so everybody can tell which of the two related documents was issued earlier. Blockchain doesn't do absolutely anything else! If a specific use case is not made easier when you can tell which of the two documents was earlier, then using blockchain is pointless.

let me rephrase: give me one good use where the majority of potential users would be OK with storing timestamped documents on a system not maintained by a trusted organization.

Ownership transfer contracts are one such example. We assume here that the signatures under a contract cannot be forged and the contract itself cannot be modified without the consent of the involved parties, but digital signatures used in (on top of) blockchain satisfy this requirement.