The reverse is also a necessity: the government approved service should not be allowed to know who and for what a proof of age is requested.
It would send the proof to you. It would not know what you do with it. I gave an example in the previous post how the identity of the user could be hidden from the service.
If the middle man government service knows when and who is requesting proof-of-age, it’s easy to de-anonymise for example users of gay porn sites.
It would be a lot easier to get that information from the ISP.