Impossible to add language switcher to new menus

I already wrote that I wasn’t able to add the language switcher to my 2nd-language homepage a few days ago and I am still struggling with it. As a workaround, I modified one additional menu I had created which had only the switcher by adding the items that belong to the 2nd-language main menu to it and used that menu as main for that language’s homepage. It works, except that the flags are not grouped like in the English homepage. What is causing this difference?

I am not able to add the language switcher to any new menu I create. The option under Customise / Menus / Settings is ticked. What is causing this dysfunction?

The page I need help with: [log in to see the link]