If you go to "All Activity" there is an RSS feed to subscribe to. You can add this to Google Reader and check it like email. All good email clients will subscribe to RSS too.
If you really want email, try an RSS-to-email service. Feed My Inbox came up top on a Google search.
There is a free plugin called Publish To Email, which I think does almost exactly what you're asking for:
Link to the answer
You just need to specify your Admins' email addresses in the list of notification addresses. I'm the author of this plugin, so if you have any requests or feedback, let me know!
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July 15, 2020: Q2A 1.8.5 released!