I think this article makes a pretty big leap in the middle. There’s really no reason that the operating system needs to be involved in the “Private” solution. It could just as easily be a website or a browser plugin. All you need is your government of choice to have some way to provide a token with whatever important bits necessary in it (“Yes this person is over 18 and a resident of WA”). You could even have third party sites/libraries that could read that token and verify what it contains.
The last third of the article is all based on that giant leap.
OIDC is designed around this capability, I could already imagine a big country flag themed “Sign in with Govt ID” button on websites like we do with other options.
Its only incompatible because there isnt a system like koreas that has basically two different id numbers. One is the general id (basic info like name and birth date) while the other is financial identity (loans, taxes and stuff). Many countries cant give their form of an id (e.g SSN) because its directly tied to something that can easily affect their life.