It does work as long as the attesting authority doesn't allow issuing a new identity (before it expires) if the old one is lost.
You (Y) generate a keypair and send your public key to the the attesting authority A, and keep your private key. You get a certificate.
You visit site b.com, and it asks for your identity, so you hash b.com|yourprivatekey. You submit the hash to b.com, along with a ZKP that you possess a private key that makes the hash work out, and that the private key corresponds to the public key in the certificate, and that the certificate has a valid signature from A.
If you break the rules of b.com, b.com bans your hash. Also, they set a hard rate limit on how many requests per hash are allowed. You could technically sell your hash and proof, but a scraper would need to buy up lots of them to do scraping.
Now the downside is that if you go to A and say your private key was compromised, or you lost control of it - the answer has to be tough luck. In reality, the certificates would expire after a while, so you could get a new hash every 6 months or something (and circumvent the bans), and if you lost the key, you'd need to wait out the expiry. The alternative is a scheme where you and A share a secret key - but then they can calculate your hash and conspire with b.com to unmask you.