By privacy-preserving, we usually mean that you get some kind of cryptographic token from an entity that knows who you are (and can attest that you are above age), and that token is anonymous, so when you use it to access a random service, that service cannot extract information about you from the token, except that you are above age.

It is possible, it just had to be implemented properly. We could complain about politicians not understanding that, of course. But if you spend 5 minutes reading complaints about age verification, you will see that nobody cares about understanding... if the people doesn't care, why would the politicians?