Admins are not notified about messages in the inbox of the moderators group. I think one of the reasons the first admin is now a moderator by default was this option
Moderators are also notified about unhandled flags and users in the review queue, as well as users who are ignored by other users.
And I think even though admins can read all messages, Discourse behaves differently when you reply to a message that was sent to a group as a member of that group. Then the individual user isn’t added to the conversation. When an admin visits the moderator inbox and replies to a message, the individual admin is added. Example from Meta: as you can see, the users who are admin and moderator were not added, but the admin was added when he handled the flag which has sent an automatic reply.
If an account is later removed from a group, they lose access to the conversations they had access to because of being a group member, but not to those where their account was added because they were not a member of the group. So pmusaraj will still be able to look at our conversation even without admin permissions, while the others will lose access once they are no longer admin or moderator.