Why they don't use zero knowledge proof? Also question for the USA constitution experts, is this considered a violation of free speech? The article is not clear on this.
Zero knowledge proof is either trivially defeated by re-using the same credentials or doesn't have useful privacy guarantees. There really isn't an in-between here for something like age verification.
The idea is that e.g. the government would give you an app that lives on your phone. When you apply for the app you provide some documents to prove your age, but you don't say anything about what sites you plan to visit. When you want to visit an age-restricted site you use the app to generate a proof that you have it, but the site doesn't learn anything more than that, and the government doesn't learn that you used the app.