Already exists in a bunch of countries. Works better in some than in others.
The issue is that you don't want everything tied to that ID. In a less than ideal world, ideally the ID would just attest that some random pseudo-ID is real. Like Webauthn, kinda.
It's kinda silly to start discussing implementation details of something that doesn't exist. Not to mention considering the alternative which is quite a bit more invasive than having an attested private pseodoidentity would be.
What is stopping them from recording the value returned to you that is then passed to the site you tried to visit? Does the data provided to the integrity checker allow for identification? Could the original vendor pass some value to use in the integrity check to prevent replay attacks, and could that value itself encode your personal information?
> Could the original vendor pass some value to use in the integrity check to prevent replay attacks, and could that value itself encode your personal information?
Well that value is most likely a cryptographic signature, a "challenge" or a combination of both. Unless there's some separate payload you can't really hide arbitrary data in hashes/signatures that would be used in such a process.
In the end "could" is a very loose word, PII as such is not really part of the process. In this current (Apple's PAT) case, the information is "you have an Apple device", can't currently hide anything else in that.