The reason it works the way it does, where you can get a picture on your TV, but only a blurry black & white image through Game Capture device, is that many TVs "share" the green component input with Composite circuitry. That is, if it detects a Composite signal, it routes it through the Composite signal circuitry to give you a good image. If it detects the "Green" Component signal, then it uses all three Red, Green, Blue Component inputs instead. The Roxio Game Capture device doesn't support that, it only supports a real Component signal.
If you can find a unit like Jim suggests above, that should work.
If not, then Roxio's VHS to DVD package supports Composite video signal input, but only for capture, not for streaming.