There are quite a few problems with that in the case of medical records.
Some of this information may be urgently needed at a time when the patient is not capable of giving informed consent for its disclosure or does not have it immediately available.
Some of the information may be vital to the future healthcare of the patient and would cause serious harm to them if it were lost.
And in more of a more morally grey area, some information might be harmful to the patient if they had it. For example, consider the implications of bluntly disclosing various mental health diagnoses to someone who doesn't fully understand what they mean and whose condition means they won't necessarily respond rationally or beneficially to the information.
In this case, having the records kept locally by exactly one organisation that is run by medical professionals who are bound by strong professional ethics seems like a reasonable policy.