You don't actually need a modded 360 to inject / extract content, you can connect a usb memory stick to the Xbox 360 and format it for use with the 360.
How to configure a USB memory stick to be used as Xbox 360 storage
Once you have done that and copied some content to the usb memory stick you can use tools such as
Party Buffalo Drive Explorer to extract the content on a PC, no modded 360 required.
You can always download title updates for games, regardless of if the game / dlc has been delisted. If you purchased some DLC in the past that is no longer on Xbox Live, you can still re-download it from your account's download history on the 360 dashboard.