If a server does not normally announce new mail messages with EXAMINE it probably not do so just because you use the IDLE command.
The entire reason why a server might not announce new mail messages with EXAMINE is if the underlying mail store requires an exclusive lock in order to read. If the server does not want a read-only session to prevent another read-write session, then it must release the exclusive lock after obtaining a snapshot of the mailbox. Having done so, it has no reasonable way to observe new mail. More modern mail stores have separate shared and exclusive locking, so this problem does not happen with these.
