In my configuration I have NO home menu in my main menu because I think its enough to have the clickable logo as home menu item. Within the mobile menu the script selects the first menu link as default where as it should be free (unselected).
Mobile menu is nothing else as HTML <select> element and it requires "starting" item (home in Your case). Since from what I understood You lacking home element it cannot work as HTML needs it to.
Simply put the home element back to Your site and tell me the url to Your site so I'll see what needs to be done to hide it on desktop.