Doesn't this mean that it's not only your hospital that sees your medical records, but... everyone who would otherwise only need your name and telephone number?
Or is there some way to restrict which party gets which data?
I don't think any of the national id services I've heard of stores all your data in a centralized place. Usually the national id service only provides identification to the service providers that request it. Each service provider (like, your bank, hospital, pension provider) will store their own data as they've always done, they just use the service to identify you.
In Finland there is centralized database of all medical records. Which makes information transfer simpler. There is ofc risk of untheorized access. But for that reason legal system exists. You get audit trail and then can prosecute or fire those who accessed information unnecessarily.
It is trade-off, but probably lot more accountable than paper records in big hospitals.
Or is there some way to restrict which party gets which data?