This isn't it. You can use a separate CA for identification and for websites (TLS). If this were the problem, they could use any existing CA for their websites and their own for identifying the user - since that doesn't involve the browser trusting the ID CA.
See RFC 5280, Section 4.2.1.3, Key Usage: https://datatracker.ietf.org/doc/html/rfc5280#section-4.2.1.3