No, you do not need a system update to play Xbox 360 games. Games are designed to run on the base console firmware. Updates add features, fix bugs, and improve security—they don't unlock game compatibility.
Settings → System → Network Settings → Test Xbox Live Connection. If an update exists, it will prompt you. Or download the offline update file (OSU1) to a USB drive.
If nothing works, contact Xbox Support—your drive may need repair.