Just say the quiet part out loud: this is absolutely going to happen, and this is why we need to fight our hardest to stop it.
Stop being distracted by and thinking about the technical points when the whole idea itself is bad, just like WEI and the other authoritarian ideas that originated with "trusted computing".
The problem with that is that the government obtained logs from the 3rd parties they might be able to compare timestamps with the timestamps of the anonymous age queries and figure out what sites some people were logging into.
Seems like a great thing then. People get annoyed, businesses that comply lose customers and money etc.
All that friction means these policies become inherently less popular regardless of anything else. While this crap work effortlessly out of the box is just outright dystopian