Zk would perfect for online age verification, but governments do not want to implement it like this. Instead they want id and face collection for mass surveillance, using age verification as an excuse.
The actual problem with ZKP is that you need a way to prevent generating thousands or millions of assertions from one ID and distributing them to whoever wants one, in a way which is undetectable and unstoppable by the government, and the only way to do that is with Google Play Integrity Protection and such.
Google Play Integrity is not a requirement, but governments think it is. All you need to avoid trivial duplication of certificates is that keys are bound to a device which is also able to perform the primitive cryptographic operations needed to construct a ZKP proof. This could be achieved using a USB dongle. Still proprietary technology, but the scope of what needs to be locked down is much, much smaller than with a solution like Play Integrity.
What stops one from selling access to that USB dongle over the internet, anonymously doing thousands of verifications per second, cardsharing[1] style?
With Google Play Integrity Protection, each such verification needs a human physically clicking buttons on the device, which makes things much harder.
https://github.com/microsoft/crescent-credentials
But of course the thing would need users in order to attract users.
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.