3rd party app... As best I know it, Windows needs a helper in the form a shell extension, just to do what it currently does.

This extension, handles making the connection from the context menu's sendto, various apps file>email, send link as... to the installed email application. That works because the various apps are configured to work with the shell extension. Like wise installed email apps are programmed to work with that extension.

I don't believe there is any way for a browser, like FF, to be programmed to act as an email client itself. Basically FF would have to incorporate a function like Gnotify into it, but also be able to deal with every webmail available. Not likely.

So, we have Gnotify. Which acts like an installed email client, as far as the shell extension is concerned, and interacts with Gmail in a similar way that a user would.

edit: Gnotify could have been designed, to operate like Outlook does. Loading only when called by a mail to: or such, But then it would, like Outlook, have to be sitting in your task bar in order to notify you of new mail arriving.