I feel you are talking about a different thing now. My point was surrounding what you initially said:
The only right way to do this, is if governments provide their citizens with an eID that any site can ask “is this person 18+?” and get an accurate answer without any other identifiable info. And if you don’t want the government to know what sites you visit, have sites route the request through a proxy.
An eID is a digital document. You yourself are proposing that sites should request people to provide a document, one that's issued by the government to you, personally. Then later you said that using a proxy prevents the government to know what you visit.
My answer was that if you are providing a government-issued document/file to the service then the government (the issuer) can know if you visit the site just by keeping track of who did they issue each document for and requesting the sites for copies of the documents. Even if the document itself does not say your name. And that's regardless of how many proxy layers you use, since there's traceability in the document. This makes you fundamentally less anonymous to the government than before (when you could have indeed used a proxy to prevent this), this makes proxies no longer a good defense.
The service does not know you, but that's not the point, what you said is that the government can't know if you visit the site, which is the one thing I disagreed with.
Oh, I see the misunderstadning.
Note that "authentication and login" does not necessarily require network communication with a government service. In fact in Europe the eIDs (eIDAS) are digital documents that use cryptography to authenticate without the need of spending resources in a government-funded public API that could be vulnerable to DDOS attacks and would be requiring reliable internet connections for all digital authentication (which might not always be an online operation). The chips are just a secure way to store the digital document and lock under hardware the actual key, making it much harder for it to be copied/replicated, but they don't require internet connection for making government-certified digital signatures with them that can be used in authentication, this is the same whether the service itself you are login into is online or offline.
In any case, in your example where actual network communication is used, it would still be possible for the government to track you regardless of proxies, because then they can store a log of the data & messages exchanged in the authentication.
They can either ask the sites to authenticate previously with the government for the use of the API (which would make sense to prevent DDOS and other abuse, for example), which would let them know immediately which site you were asking login for (in a much more direct way than with "documents"), or simply provide a token to the site as result of the user authentication (which is a common practice anyway, most authentication systems work through tokens) and later at any given time in the future ask the sites to provide back which tokens are linked to each account on the site (just like I was saying before with the "documents" example) so the government can map each token with each individual person and know which users of that site correspond to which individuals.