IIUC the only thing keeping bitcoin's blockchain trustworthy is millions of people mining new coins and therefore distributively verifying the blockchain. Who would be doing that for your medical records and what would their incentive be to do it?
But, for medical records there is no such incentive. Therefore anyone can easily change the records or add new ones and claim everyone else who has a shorter chain has the wrong chain (which will be like no one since there is no incentive to mine).
So I'm probably just informed how blockchain is supposed to help here. Without the distributed trust there's no plus to blockchain. And without the incentive to mine that generates millions of miners there is way to have the distributed trust.
I'm happy to be wrong but I haven't see an explanation how this issue is solved for all these non virtual currency uses cases.
Ethereum would be easier to understand and see what's going on because we have a reasonably readable contract language. You'd still likely need to store larger blobs of data using an external immutable data store, and those blobs wouldn't be automatically distributed and backed up. But you could pay for distributed backups in an automated way using ether. Lots of people working on this last part, which drastically reduces the potential size of the blockchain because it will only be afordable to log blob references and numerical values.