The EyeTV Hybrid does not support HD or widescreen input from anything other than digital cable or antenna broadcasts, so the quality from a game console would indeed be affected. See
this article on Elgato's website for more info.
I don't know about the newer EyeTV Hybrid or the Hauppauge, but I've tried this with the EyeTV 250 (a slightly older model) and it also introduces a second or two of lag, even in the EyeTV's "game mode".