No it absolutely does not. I think you're confusing CT with OCSP, which is the browser checking to see if a certificate has been revoked, which died a death not just because it was incredibly slow and unreliable (CA's OCSP server fell over enough that OCSP failed open), but more critically because it was an incredibly invasive privacy problem.
CT does not require that, that was part of the explicit design goal of CT. To be trusted by a browser a CA has to publish every certificate they issue to multiple CT logs. When the certificate is published to a CT log, the log provides a signed confirmation, and that is included in the final certificate.
Every browser has a set of CT logs that they trust. When they perform the trust evaluation on a certificate they verify the certificate is correctly signed by the CA, and then they go through the included CT annotations and find the annotations from logs that the browser trusts and verify that those annotations are signed with the respective log's public key.
At no point does the browser perform any other network requests (again, a problem with OCSP is that the additional network access during TLS handshakes was incredibly slow and unreliable - if CT required this it would require _more_ fallible network loads). That makes CT faster, and resolves the privacy hole OCSP forced.