One problem with private age verification is that because each verification cannot be traced back to a user, it is hard to prevent abuse like credential sharing. Imagine how a single stolen credential can be used by any number of users because the verification step kept the credential private.
One method would be to use the same key that you use to hold some cryptocurrency, so if you share then you risk losing a bond.
Of course it's not ideal to make everybody hold crypto just to use online services, but maybe we can approximate that in other ways. Say, have the private data include name/SSN/DOB and maybe a credit card number, require the user to enter that stuff (or have browser do it), prover checks that it's all correct. Combine that with a challenge/response so proofs can't be reused. User can't share credentials without risking identity theft. Downside is more openings for local malware to succeed in identity theft, but maybe that's better than sending full credentials to big juicy central locations.
A third option would be to give everyone a hardware key that's hard to copy, but that would get expensive.
I think the best idea is to just skip age verification and keep the good ol' internet we've enjoyed for decades.
I assume they are solving this with secure enclaves creating one-time signatures.