It would be the TV, modern ones seem to not care about quality regarding older resolutions or obsolete signal types. I wouldn't know exactly why in technical terms the TV is having these issues, but running the signal through a second device seems to affect how it's being filtered.

If you want an HDTV that can display the old consoles correctly, you'll probably need to look for an older model rather than a newer one. Few people care about running those old connections now so manufacturers will cheap out with the circuitry in those areas. Lots of modern TVs don't even have built in TV tuners anymore, it's really disappointing how poor the current selection of modern TVs is.