Add a 'virtual page' from plugin

Yea – ideally, when someone visits: site.com/my-plugins-public-page, the site will use the theme’s normal page template and allow me to put my own content inside of it.

I don’t want said page to exist in the dashboard, database, pages widgets etc.

I found this http://wordpress.stackexchange.com/questions/9870/how-do-you-create-a-virtual-page-in-wordpress (the answer with 10 upvotes)

which is letting me execute my plugin’s php file from the URL I want – which is great, so now I really just need to figure out how to wrap my desired output inside the page template… which is proving difficult