While the question of "is it actually possible to do this in a privacy preserving way?" is certainly interesting, was there ever a _single_ occasion where a government had the option of doing something in a privacy preserving way, when a non-privacy preserving way was also possible? Politicians would absolutely kill for the idea of unmasking dissenters on internet forums. Even if the option is a possibility, they are deliberately not going to implement it.
I didn't know about e-IDs in other countries, but in Scandinavia (at least in Norway and Sweden, but I know the same system is used in Denmark as well) they are very much tied to your personal number which uniquely identifies you. Healthcare data is also not encrypted.
Well the e-ID is an ID, so to the government it's tied to a person. But I know that in multiple countries it's possible to use the e-ID to only share the information necessary with the receiver in a way that the government cannot track. Typically, share only the fact that you are 18+ without sharing your name or birthday, and without the government being able to track where you shared that fact.
This is privacy-preserving and modern.
> was there ever a _single_ occasion
I don't know where you live, but in my case, many. Beginning with the fact that I can buy groceries with cash.