> If you can follow that logic

I can't, because it's not logical. You want a credential that provides exactly one bit of information with certainty. It's not at all clear that something like that exists or if it did, how you would prevent multiple people from sharing the same credential if it was also actually possible to use it anonymously.

> It can represent "this anon identity is a PhD in physics", "this one is a lawyer with 5 years experience in criminal law",

You're saying the credential can leak information about the user. You don't need many of these bits of information to de-anonymize someone.