The purpose would be to reduce "email notification distraction" in very simple way (e.g. for when I'm extra busy for a little while and do not want even the usual minor distractions). Shutting down the application seems like overkill, especially since it slows me down even more later when I do want to check if I received any email.
Ideally, it would be a toolbar button that toggles notification settings, with an added option to have it activate anytime I simply minimize the Thunderbird window (and restoring them when the window is re-shown). I have searched but have not been able to find an add-on that does anything similar, but since email is a well known source of distraction I thought I'd throw it out there to see if anyone has heard of such a thing, or knows if it would be easy to create such an add-on.
The settings I'm referring too are the Options > General > [Show and alert | Play a sound] check boxes. The little envelope icon in the system tray is less intrusive, but it could be optional to turn that off too. One of the keys would be for Thunderbird to quickly show any new notifications when they are turned back on.