The component cables are the only way to get 480p on a SDTV from the Xbox. All the other cable solutions will only do 480i. And this is also the only way to get 720p for the handful of games that support it.
I'll have to agree with KnArFnB that it is probably your switch box. If you have no problems when plugging directly into the TV then that basically confirms it. I am currently using the monster cables instead of an HD box (the "official" Xbox Component cables) and have been very happy with them. They have a bit more shielding that provides a slightly crisper image. The colors will also have a bit more depth as well as black being black instead of grey.
Also I would suggest having the component cables go directly into your TV instead of through your system selector. I am going to guess that you have a fairly inexpensive component switchter, which means it will consist of cheap and unshielded parts. This will introduce a level of signal interference that will degrade the signal quite a bit. The cables between the TV and the selector will also reduce the quality. Having monster cables going into this type of setup is kind of a waste because any benefits you get out of the expensive monster cables will be negatated by the cheap system switch and extra cable.
OR.. you can go with an expensive shielded switch. I've been extremely happy with the one in my sig. LINK Even with it I still have my 360 going directly into the 2nd set of Component plugs on the back of the TV. Everything else that uses component is routed through the switcher. The cables from the switcher to the TV were quite spendy as well.
But yeah, my response below was basically useless. Sorry about that. I somehow read your post thinking you were not wanting to have to unplug the cables everytime you wanted to use another system.
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