Greetings!
We have the same experiences and we are using Google Apps for Edu to handle our university email. Prior to switching to Google, about 4 years ago when were using another in-house email, the faculty received their emails from Moodle.
I mention to faculty that they can go back and check messages/postings that have sent by going to the News/Announcements forum and/or to Messages. I found that they like knowing they can do this and its all in one place rather than having to search through their email. Of course users' needs and preferences will vary. There maybe something we can configure in our Google domain gmail settings and/or in Moodle to capture these emails but have not looked into that.
Best!