Sorry for the late update, but I did manage to get the better S-Video cable wired to the cheapo connector and the picture looks great. The problem is that I've been too busy to properly solder and heat-shrink insulate the connections, so it's currently in a rat's nest of hand-twisted wires. The only thing I've done in the past couple months is to impulse buy some heat shrink tubing from Fry's while I was there to buy a new wifi router.
I've attached a picture below. In the process of writing an explanation of the current state of things, I just noticed that the hand twists have come loose again - damn!
At any rate, the thick black cable is the true S-Video cable, with the blue and red colors being the outer insulation around the coax-shielded signal wires. Below that are some horizontal-pointing thin black and brown wires, which are the unshielded S-Video signal wires of the cheapo cable.
I really need to get this cleaned up so that I don't have to repair it every time my 3-year old daughter wants to play "the monkey game" (Donkey Kong 64 - currently inserted) or "Link" (Ocarina of Time) or "Yoshi" (Yoshi's Story). I still haven't decided if I want to reconnect the composite wire; it's kind of a don't-care since I have one or two official Nintendo composite A/V cables already.UPDATE:
Finally got out the soldering iron and finished it up. Turns out the heat shrink tubing was too small to be useful, so I just used electrical tape to cover the connections after soldering. I also taped the big cables together to try to provide strain relief in case small hands tug on it, as without it the stress would be on the small wires and/or solder connections.