My vision of the program: icon in tray, left click on icon opens a window with shutdown, reboot, etc. buttons.
The new Gnome specification advises to use notifications instead.
StatusIcon >! (Deprecated since version 3.14: Use notifications)
Unfortunately, I didn't find how Notify.Notification can be displayed in the tray. The only thing that I found is AppIndicator3.Indicator, but its constructor can only take a Gtk.Menu, which is unsuitable.
Which classes may I use?
I had used GtkStatusIcon due to comments