You've explained well, but I can't say I have a clue as to why that would be. The display on certain pages is completely controlled by Joomla itself. Our modules don't enable that functionality at all. The only time I've heard of issues like that is if you're trying to say it can only show on a certain menu item and that menu item is actually an alias to another menu item. You should always set it to display on the original menu item, not any aliases for it.
Beyond that, if you still can't get it, I'd recommend enabling all but one menu item to see if it works. Then, if so, start unchecking large amounts of menu items to see where it starts disappearing. Then, debugging why certain menu items have to be enabled.
Hope that helps,
Alex