Can't you just put a middle man on there then? Get a non-profit organisation like Mozilla to ask the govt. on behalf of the user.
The organisation asks the govt, and gives back a signed token.
The the only thing the government knows is that an age verification was requested. Once verification has been done once for one site, it can be used for future verifications.
The middle man in this scenario can mask the URL that is requesting age verification, but what's to stop the government compelling traffic logs from the middle man?
Nothing more than what prevents them from getting logs from your ISP about the sites you visit after verification. In ideal countries they need a court order for that, in less ideal ones they just scoop up the logs preemptively.
The organisation asks the govt, and gives back a signed token.
The the only thing the government knows is that an age verification was requested. Once verification has been done once for one site, it can be used for future verifications.