As I understand it, when you add a web-part or app-part you are creating an embedded version of an existing app rather than a brand new app. So while they are very functional by themselves, in many cases you can't add an web/app-part to a page if that app does not already exist in the same site/sub-site.
If you create a Promoted Links list (even if you hide it) then you can add a app/web-part to your page. One thing to note though is that the web-part/app-part that you add will be linked to the source app. So if you plan on adding several of them to various pages, a good practice would be to create views in the original Promoted Links list for what you want to show on each page you add the web-part/app-part to. That way you can use the one list for everywhere.