"Mailing Event: Subscribe to mailing list" action only possible for public groups
It seems to me that for the "Mailing Event: Subscribe to mailing list" action the group you to subscribe to needs to be visible publicly.
At the same the comment in the code of the "Mailing Event: Subscribe to mailing list" action suggests that it should be possible even for non-public groups.
I'm running CiviCRM 5.59.2, and Action Provider: Version 1.124