How to find the app that causes crashes? Where should I begin the hunt for the cause of crashes / reboots when logging apps do not work anymore?
As can be seen in this video from the Google I/O conference, in Jelly Bean applications can no longer read the system log for other applications, they can only read their own log. This has broken most (all?) system log readers, here's a blog post from a developer of such an application talking about how it has affected his application.

The only way to have an app read other apps system log entries when using Jelly Bean is to root the phone.

Although other apps can't read the log files it is apparently possible to do it from the PC, try giving LogShark a go.