How to provide email notifications to admins for *all* comments?
I see that you've already found the Comment Notifications comparison page over at Drupal.org - but also that you say you have not found the module you're looking for there.

However, from the description of your requirements, it sounds like the Notify module is pretty close.

Since it uses a digest format and depends on cron, it will not give your site admins instant notifications of new content, but otherwise, it matches your description.  Most important, It will send out notifications about all new comments, all new nodes, or both.

Disclosure: I am a co-maintainer.