There's no merging or rebuilding. You just nuke the bad log, and certificates continue to work because they are present in other logs. (If all the logs used by a certificate go bad, then you'll have a bad time. This hasn't happened in 5 years of mandatory CT.)

The log operator may choose to stand up a replacement log. The new log will have a different key, URL, and name from the old log. It's for all intents and purposes a completely different log.