The share buttons and navigation chrome certainly can be put in a PDF, they're just incredibly uncommon. I've even played video games that were distributed as PDF files.
The ones I played were adventure games, basically a bunch of areas (implemented as pages) linked by buttons, with some extra logic thrown in. If you can imagine the kind of scripting capabilities you'd need to run a PowerPoint style presentation from a PDF file, and the kind of scripting you'd need to make sophisticated interactive forms, you're on the write track.
Unfortunately I don't know what the games I played were called.