Question, does your LCD monitor switch to a "TV Mode" when displaying S-Video or Component content? Does it literally take any video feed and expand them to 1600x1200 no-matter-what?
I imagine you're looking into a GameCube system which lacks Component Video-output, with your best option being an S-Video connection. S-Video isn't bad, and GameCube content (~640x480) shouldn't look as aweful as the N64 content (~320x240) did, because the N64 content must be upscaled to a greater extent.
Try hooking up a DVD player to the LCD via S-Video. If you can live with what you see, then a GameCube image should be fine as well; they essentially have the same resolution.