Put whatever keybind for switching input language I can't set CMD + Shift as a keybind for switching between input languages of the keyboard. I had to settle down with Ctrl + Space. Why does it allow me to set only some specific key binds?

Is there a way to go around this and set specifically L cmd + L shift ?
The command as well as the shift key (and some other keys) are modifier keys. You have to add at least one normal key (like A-Z, 0-9, Fn*, arrows etc) to define a proper shortcut.

If you want to change this behavior you have to install a third-party tool like Karabiner.