After some checking, it seems to be an issue with the free Heroku instance they are running.
Amazing what they squeeze out of a single process architecture.
I think they could move it to a more powerful machine and the remaining issues would be fixed. It wouldn't be as big of a flex as running on an ancient Intel CPU though.
Edit: I got "We're having some trouble serving your request. Sorry!" while trying to send this comment :)
The front page is ~7 kilobytes on reload, and ~20 on first load - such a good example of "less is more".