I check mine once or twice a month manually, but it is pretty trivial to monitor it automatically as well, e.g. there are APIs for crt.sh or they even offer direct public read access to their database.
I believe certificate users should remain responsble for their own monitoring. Alerting as you say would be very annoying since you couldn't preemptively replace certs without getting alerted unecesarily, and it would divert Let's Encrypt developer resources away from more useful projects.