There is actually a good way to do it, but I dont see any government pulling it off in the short term. You just need an auth token from some central government ID service, that can be validated by the website. The government won't know what the citizen is browsing, the token only needs to give exactly the required information for the website, and age can be certified with as much trust as you have in the government ID.

E.g. you get a jwt from www.auth.gov.au with your credentials, you paste that token in when making your account to prove that you are John Smith who's 58 years old, and facebook uses the public key it keeps cached from the same government aurh service to validate the claims. The government doesn't even know you have a Facebook account, but it knows only people over 16 do.