
https://www.routledge.com/Games-Puzzles-and-Computation/Hear...
It has a huge number of examples of how to build similar gadgets using lots of different kinds of games.
It's an interesting read. Not too mathy. They make clause gadgets in the same way as the example here for Super Mario Brothers and use that to prove things.
Edit: Realized the author of this book is actually one of the authors of this paper. So, yeah, if you found this article/paper interesting be sure to pick up the book.