I think, the idea is the distributed ledger, which allows you to store the information not with a centralised authority, but yourself. Blockchain would not necessarily be a good technological approach, but a distributed ledger, would be.
In general the healthcare system can't rely on patients to store their own data. Patients lose devices, forget their passwords, get hacked, etc. Certainly we should make it easier for patients to obtain copies of their medical records but it just isn't feasible to put the patient at the center of the data flow.