Only Administrator can send announcements and alarms from the Alerts portlet via email transport.
A User is associated to the Community Super Administrator custom role - this custom role as the permissions to Assign Members and to Manage Announcements on the TEST role.
But when you create a new Alarm setting TEST role as distribution scope, no EMAIL is sent.
Note that all other setting are ok, including the destination account that decided to receive the Alarms via EMAIL transport.
In fact, the Administrator is able to start the email distribution.
Please, see the screenshot attached for the permission on the TEST role of the Community Super Administrator custom role.