Creating a page that does not show up in the frontend navigation bar

Auto show a page in a navigation bar is not a standard feature of WordPress. It is a specific feature of theme or plugins. If the theme you are using has this feature and you don’t want it you will need to create a child theme and modify this behaviour.

Anyway, what you have described is really better done with a custom post type. Think about it. You are using a page (a post type) to embed another type of content. WordPress allows you to define custom types of content, called custom post type. Use it, it is much much much better: custom templates, managament UI, searh, queries API, meta data API, object chache, permlinks, easy import and export, and so on.