And as a follow up to my previous point, now that I'm at my PC and don't have to type on my phone:
Even if we made certificates in a way that can't be shared, for example by locking the private key inside a physical device (like a digital passport), we have solved nothing.
Your certificate would immediately become your digital fingerprint that will be stored with your account (to find duplicates) and can be tracked across websites as soon as a database gets leaked or the sites' owners sell your data to advertisers (when would that ever happen?). While that fingerprint alone doesn't say anything about you except your age, it makes it trivial to aggregate your activity across the whole internet. Ever bought something on a site that requires age verification? Congratulations, your certificate is now tied to an address. Shared a selfie somewhere? Your certificate has a face. Even without personal data directly in the certificate, it would be a privacy nightmare and exactly what the EU GDPR tried (and failed) to prevent.
The next step would be to find a mechanism that creates single use certificates every time you need one. But you can't do that locally, because the certificates still need to be signed (and revocable) by a trusted authority. So maybe you need to send a certificate signing request to a government server every time you sign up for something. That could work for some use cases but requires expensive infrastructure that is never allowed to fail even for a few minutes or it would cause chaos.
... and now I've noticed your exact wording, implying that sites would forward the users' certificates to the authority to be verified. That's a big no-no. A site may never ever acknowledge to an authority that it has seen a specific certificate. The authority necessarily knows who the owner of that certificate is and even if they don't tell the website, the authority itself can keep track of every citizen. "On date X, PornHub asked us to verify the age for certificate ABCDEF which we know belongs to John Doe from Somesmalltown" is not something I would want to be stored on a government server.
And this is all still assuming that the infrastructure for this would be implemented according to modern standards without security-critical shortcuts. If you have any hope that will ever happen, I recommend you click through https://media.ccc.de/ and watch some talks about government IT fails. Many are available in English.