The launch log you included isn’t showing a problem yet, other than you not being patient enough. OVH can take a really long time sometimes when I use it from the United States. I have no idea why. It isn’t consistent in a lot of ways that make it hard to ascribe the issue. It isn’t consistent as some launches from some repos are fine but hitting the same one later in the day sometimes can be slow. (More about OVH below.)
I found changing to Google worked for launching your repo on MyBinder.
However, Shiny still fails to work. (But right not Shiny example isn’t working either [see below].) And I think there is a memory limitation issue so that this won’t work while Gesis is being odd today. I opened RStudio from the session and tried running your app.R code in it and the console seems to crash. (R Session Error: crashed the R session due to unexpected crash and lost the workspace.) I figured it was a memory issue as Google and OVH are much more limited than the offering Gesis provides. I may be wrong though. It was just a guess given what I saw.
Returning to trying from OVH:
So right after a very launch log showing and length and things very similar to what is in your post, I had a successful launch on OVH. It took like at least eight or more minutes to work, but it did. Your app of course still fails to initialize just as it does at Google. But OVH will work for a launch of a session if you keep waiting. I just switch to lab after it launched on OVH and then tried running your app.R code in RStudio again and again it crashed the R session due to unexpected crash and lost the workspace.
More disturbing to me for the prospect of getting your Shiny app working quickly again is that the current Shiny example from the binder-examples/r repo here fails to work. While it launches fine. It has errors on it so that the Shiny dashboard doesn’t work. I don’t see any report of an issue with it yet. Although this may be completely unrelated to the issues above. Maybe it is is just a coincidence I stumbled upon it while trying to compare and contrast Shiny launches.