Ben Miller points out a gap in functionality:
Call to action for Microsoft. Contained Availability Groups came out in SQL 2022 and they definitely have their use. But there were some artifacts left behind that need some fixing. Namely when you use DBMail while in the Availability Group jobs or operations. Let’s see what there is left.
First, here is the link to the Feedback Item that is out there for voting to get Microsoft to fix this issue. There has already been an issue fixed with the msdb proc to activate dbmail in a Contained AG ([dbo].[sp_sysmail_activate]).
Read on to learn more about the issue.