Website generates a random nounce, the government offers to sign the json { over_18: true, nounce: [inserted_here] }. That signing is coupled to you being signed in, or scanning your passport via nfc or whatever

That significantly reduces the leaked information: the service doesn't receive any information who you are, and the government doesn't know what service you use. They can collude, but that does not reveal more to them than most current age-verification methods. Children can get another adult to sign their request, but that's a working attack on nearly any age verification (including in-person purchases)

Of course this doesn't replace parenting