Simple answer.... NO
What you want to do is not possible at all. To be able to any software on an xbox other than the intended xbox dash, you need to have a modded system, whether that be a chip or a TSOP flash, etc. An unmodded system will have the original BIOS on it, thus making it impossible to run any other programs. I think you are misinterpreting the modding process. The modchip/exploit is not done so that you can load things onto your hard drive, it is done so that your system can be flashed with another BIOS. That BIOS allows you to alter the files on your hard drive, not the other way around.
If you put your hard drive into a non-modded system, it will (depending on your file structure) run the xbox dash and act as though no files are on the drive other than what should be, or it will just not boot up correctly at all.
Even if it does load the xbox dash, it will crash eventually, and you will not be able to run anything from the drive.
You need to mod your system in some way to run the files on that hard drive... no way around it.