The solution is really, stunningly simple:

Your gov issues official documents about you (driving license, passport, id cards...). They know your age.

Your gov is also a trustworthy institution since all those cited above are official documents that anyone, anywhere will accept as valid.

So here's the solution: the gov creates a digital certificate in which the only stored data is your age, or even less: your adult state (as a boolean; if over 18 = TRUE).

The gov issues the cert on demand to any person after presenting any valid ID to prove who you are (it can be done online, with only the id verification being done in person). The cert is bound to your device, and if you change phone, you must migrate it so you can't have it in two devices.

Since the issuer is a trusted authority, the cert can be used as a proof of age in any site needing it as the only thing they need is to read the cert and confirm the auth of the issuer.

And as the cert is only a boolean status saying if you are underage or adult, there is no privacy concerns as the one checking your age won't know anything else about you.

There, you just solved a "huge" problem in a simple way and with no privacy concerns.