What is preventing ORA-00376 errors appearing in the alert log?
Oracle provides the following information for this error message:

ORA-00376 file string cannot be read at this time
Cause: An attempt was made to read from a file that is not readable. Most likely the file is offline.
Action: Check the state of the file. Bring it online.

Their logic for not including this error message into the alert log is related to the fact that the alert log is to show messages received by the background processes that make up the instance. User session error messages could indicate larger problems with the instance, but all user session error messages are not indicators of instance problems. (ORA-00001, for example)