"you could also try to run the automated backup outside the Moodle cron with a bash script that is started by the Linux cron"
Thanks for reminding me. This one is a poorly maintained Moodle instance which I am trying to stream line. Cron task was one of the things that were neglected. In fact I started doing the "automated backup manually" through the cronjob of www-data. Then for some reason switched back to the cron of Moodle. Yep, the PHP CLI backup script is much faster.
BTW, a Bash script is not necessary, just '/path/to/php /path/to/moodle/admin/cli/automated_backups.php' in the cronjob is sufficient.