I just wanted to make sure that no message is actually sent. It could also be that no one notices because the notification level of the moderator inbox is set so that it causes little (in this case, perhaps too little) noise.
I think it could work to use data-explorer and automation to send a message or post in a topic in the staff category when @system silences a user.
Something like:
SELECT target_user_id, details, updated_at
FROM user_histories
WHERE action = 30
AND acting_user_id = -1
AND updated_at >= NOW() - INTERVAL '1 hour'
And a “Schedule a PM with Data Explorer results” or a “Schedule a post in a topic with Data Explorer results” automation that runs every hour