Local mail is never checked

It is really, really not the terminal emulator's job to set basic environment variables.

It should either be done by the environment in which gnome-terminal is started (e.g. systemd --user), so that the terminal just transparently passes this on; or should be done by the shell initialization files, as in your workaround.

Sorry, I didn't know where to open this.

Anyway my workaround... does not really work.
It only works if I receive a new mail while gnome-terminal is running : in this case I will have "You have new mail" at next command line prompt.

If I receive new mail while gnome-terminal is not running, THEN I launch gnome-terminal : no notification.

But you are right, I should move to askubuntu.

I'd take one step back and ask the question: Why would anyone one a graphical system need to run a terminal emulator in order to get notified about new mail? What if someone uses all kinds of other applications (web browser, document editor, photo viewer, music player...), just not a terminal, why wouldn't they be notified? The right approach is not to fix whatever happens inside the terminal emulator, the right approach is to have a graphical indicator on the desktop.

Well, my approach was very simple : mail notification in terminal has worked for decades, I used in the 2000s, so why not use it again if it suits me ?
It is not mandatory to "reinvent the wheel" for an old need.

Anyway, I already tried to use /var/mail/$USER in my graphical email client (Thunderbird) : trust me it's very buggy. (Mail is only checked at start, sometimes "read-only", sometimes it becomes duplicated)

But I will gladly examine any other graphical solution, provided it's (almost) as lightweight as the terminal one :)

I'm just piping in here to say that the issue is based on how the login(1) program is called during the authorization process. This is beyond the scope of gui terminals and is even once more removed from the shell.

I've outlined a more in depth answer here: https://askubuntu.com/a/1278257/879293

Also, I've provided a bash function that you can add to your .profile (assuming you use 'login shell' for your terminals) that will give you something like the behavior that you (and me as well) were seeking relative to this thread.

Best regards.