This would be impractical in meatspace, but works perfectly fine on the internet.
The data stays with them[1].
I think you grossly underplay the current practices.
[1] there's no hard, irrefutable proof companies like Persona (intimately connected with known law abusers, ie US government) keep their promises or obey the law.
[edit] I did a little reading and it sounds like the company does not query the government with your ID. You get the cryptographic ID from the government, and present it to a company who is able to verify its validity directly. My source is mostly this: https://www.eff.org/deeplinks/2025/04/age-verification-europ...