He says that because it's not clear how your answer relates to his question.
Your second post is more helpful but feels a bit backwards as an explanation (coming from "use combostrap" instead of coming from @herrdeh's question).
So let me try to rephrase your answer for @herrdeh.
Providing a sidebar, visited pages, etc. are all mechanisms provided by the template used in the wiki. The export mechanism however only exports the (rendered) content of a single page. The sidebar for example is a different page alltogether, so it's not part of that export.
So to do what you want (a layout with certain functionality, but no navigational elements) you would need to create your own, custom template. You also need a way to only load this custom template when your wiki is embedded in your other site, but not when you're using the wiki directly. To achieve that, plugins might exist or can be written.
@gerardnico suggests that his combostrap project might be able to achieve what you want in the future (though I am not sure it does cover your usecase exactly, since you want different layouts based on some environment instead of based on the currently viewed page).