The problem is that it leaves a paper trail.

Grace also knows what number n got verified, and the identity of the user n. Later, the website can ask the age-verifying service who user n actually was. It requires that the age-verifying service cooperates with the website, though, but this could be mandated by law, which would create a single point of (privacy) failure.

PS: i love your writing style. It’s so simple and clear :)

Cryptography is a really complicated subject. You managed to express it very easily understandable.