While I appreciate the zero-knowledge proofs is considered, how the hell did no one in charge of the app design think of this? It's is literally the first question I asked when I first heard about this app. You go to the app in a store to buy alcohol, you're asked to verify your age, but that's not what you're doing. Your simply showing the store that you have a phone, with and app, which was configured by some over 18 (maybe).
Honestly I don't think it's possible to verify that you're over 18 without also providing something like a photo ID (and even that is error prone).
You can probably do something online, where the website or app does some back channel communication to a server that verifies a token. Even that is going to have issues. You could add a "List of sites that has verified your age" option where you can revoke the verification, in case your nephew borrows your phone.
They are going to implement this and it will be "good enough", but I don't see this being 100% secure or correct.