Monitoring CT lets you verify that somebody renewed the certificate, but it doesn't verify they actually installed the replacement correctly.

My employer (Kynd.io) currently monitors public web sites for customers so we can flag e.g. "Hey this site cert expires in a week! If it's dead probably just switch it off, otherwise renew the certificate" and we're in the process of integrating CT but mostly so we can say "You already have a newer cert but need to go install it" in our How To Fix instructions.