DVI is either Digital if DVI-D, Analog if DVI-A or either/both if DVI-I.
Sat boxes with DVI output have them enabled. Some have HDCP while others do not. For the sake of completely answering any unasked questions, DVI is a subset of HDMI and so you can convert one connector to the other without video problems (HDMI can also carry audio while DVI cannot).
The analog video used in DVI-A (and the analog portion of DVI-I) is RGBHV, while component is Ypbpr or YCrCb. You would need a transcoder to convert from one to the other. This process will not improve your picture quality. At best it wouldn't alter it, but if you get a lower cost transcoder ($200) you will slightly degrade the video.
It will be cheaper and give you better video quality to make the extra run. I understand all about the problems with video switching, as I have 6 (yes, SIX) HD/Progressive Scan NTSC sources to run to my display.