the issue arises when you are unconcious and there are certain risks to treating you the staff needs to know about. Maybe your throat closes for benedryl but you're showing obvious signs of an allergy? Maybe you have a pre-existing condition which gives you symptoms that signal a stroke? the risk is always going to be there
They don't need my full medical history for that. Just a highlights or "in case of emergency" summary.
Humans dealt with these issues just fine before records were centralized. And still do e.g. when I travel to another jurisdiction. Being an engaged and proactive actor in my recordkeeping actually improves my posture.
I deal with the same issues when it comes to estate planning (e.g. what happens if I become unable to make decisions for myself). There are tactics and contingencies available (like trusted individuals, powers of attorney, etc) and I don't need my full financial background to be openly available to thousands of participants in the system to accommodate. Society and the concept of trust did not evolve over millions of years to end up becoming nothing more than an individual and a computer somewhere in the cloud.