What can I do with the MySQL client's system
command? Why does it exist?
The most productive use I've seen is to look around the file system (e.g., to remember the file name you want to SOURCE or LOAD FILE).
mysql> system ls /tmp
backup.sql
mysql> source /tmp/backup.sql
Query OK, 1 row affected (0.02 sec)
mysql>
As near as I can tell, you can't pass a query result to the STDIN of a command, you can't pass STDOUT of a command into a MySQL query.. it just doesn't seem widely useful.