Despite saying that, I see numerous ways in which you didn’t follow the guide repo.
Once users are more familiar with how MyBinder works and the requirements for various abilities, they can start with a blank repository and convert it to be Binder-ready. As a Binder initiate, the way to go about using the example repositories is to use them, in your case it would be the example R repo that had Jupyter + R, RStudio and RShiny, to make a new repo using the Use this template button, and then gradually convert it to their custom code and see when things break. For example, your repo doesn’t even launch RStudio properly right now. Indeed, when you click on New under the Jupyter Dashboard, you should see R as a kernel choice below Python 3. (Compare what happens when you launch from the example and then click on the Jupyter logo in the upper left to go to the Jupyter Dashboard. The New drop-down menu will be over on the left above the list of files.) You aren’t using the two file approach for your Shiny app, see here and the discussion linked from there. The list of detailed discrepancies goes on and contributes to the symptoms you are seeing.
R not being listed on the New drop-down menu in the Jupyter dashboard from lanches from your repo is a big red-flag. If R isn’t available, you cannot expect much else R-based to work, and so that is a good place to start.
It would probably also help if you searched the terms rstudio and, separately shiny, on this forum and looked at what was discussed and examine repos linked.