Can it really be the case that there is an option to be notified of a public Facebook comment but not when someone sends me a private message via Messenger?
Messenger will not send email notifications, but it does provide notifications whenever you receive a message.
Facebook and Messenger are different applications. Much like Messenger, none of the instant messaging apps I use support sending email notifications.
The 'comments' example is apples-to-oranges given that:
- Comments are not messages on Facebook
- Comments are not a feature of the Messenger app
- The two apps have distinct notification systems. To underscore this, Messenger is not an option for Facebook notifications (comments included).
Messenger
Messenger supports Push notifications in any of its apps including the web application. It does not support Email or SMS notifications.
See Desktop web section in Messenger features > Notifications
Facebook
Facebook supports Push, Email, and SMS notifications for "messages you've received directly in the Facebook App, and not through Messenger." It does not support notifications in Messenger
See the Messaging section in Facebook Settings > Notifications