Since it launches properly from the terminal, you can
Option 1) simply create a new shortcut for it as described here. or
Option 2) repair the existing shortcut at usr/share/applications/unity-control-center.desktop which on my system looks like this:
[Desktop Entry]
Name=System Settings
Icon=preferences-system
Exec=unity-control-center --overview
Terminal=false
Type=Application
StartupNotify=true
Categories=GNOME;GTK;System;
OnlyShowIn=Unity;
Keywords=Preferences;Settings;
X-Ubuntu-Gettext-Domain=unity-control-center
Option 3) reinstall the package with:
sudo apt-get --reinstall install unity-control-center
Any of these approaches should work.