Finally, when started on April 12 it finished its backup run on May 18. I received this mail today:
Automated backup pending; 0
Some of your courses weren't saved!!
Please take a look at your backup logs in:
That's not a really good score... about 25% of the courses actually make it through the automated backup. I will have a further look, but if this is not easily fixable I'm forgetting about the automated backups in Moodle and will restore a server backup instead. That seems to be way less time consuming.