From what I understood, the "attester" is a remote server, which signs the attestation with its own key, after somehow verifying that the browser and operating system and drivers and machine is not running any code that this remote server does not completely trust. That key can be used at most to identify the remote server, which is supposedly shared by a wide number of devices.
Yes, this means that your browser depends on having a working connection to that remote server for every attestation it makes, and that if that remote server colludes with the web page (or is compromised), it can leak your identity.
So you're at the complete mercy of the attester (and of whatever deals it made with the sites) but the sites technically can't use the token to track you. Privacy!!!