If age verification is really the intent then it ought to be possible to develop a service these websites can call into that gives some kind of zero-knowledge age check. The age check service doesn't need to know the identity of the service that's asking, and the requesting service doesn't need to know the identity of the person whose age they're checking. You'd authenticate on a site that only knows someone's doing an age check, and the verifying site would just get a token indicating that the age check was successful.

Am I missing some reason why this wouldn't be possible? It seems to be a problem ripe for zero-knowledge solutions.

If it is possible, there's really no need for an age check requirement to involve disclosing your identity to the site you're visiting, or to disclose your viewing habits to anyone. And if governments or lobbyists are pushing for everyone to upload their full identity to web sites, it suggests either they're ignorant or their motives aren't what they claim.