Why would I use the MySQL "system" command?
There is actually a far more useful application for the SYSTEM command than simply executing ls to see the current directory contents:

As mentioned in the docs (and elaborated on by a comment), this is way to make backups:

echo "FLUSH TABLES WITH READ LOCK; SYSTEM snapshot.sh; UNLOCK TABLES;" | mysql


snapshot.sh contains code to make an atomic snapshot of the mysql tables, using LVM, ZFS or btrfs capabilites.
You can't split this up into multiple commands from the shell script, because as soon as the mysql session is closed, the table lock is released.