Get a Wii that plays GC games and purchase a Wii component cable for it. Most major NTSC Gamecube games support 480P through component cables which look MUCH better on HDTV's than through composite(the standard red, white and yellow leads that come stock with all consoles) as component allows a higher resolution and truer picture quality. It would actually be cheaper to get a Wii console AND Wii component cable than it would be to get a Gamecube component cable because they are super rare and people want $250+ for the component cable.
Important note: Most PAL Gamecube games had the 480P support stripped out even though the early PAL Gamecube consoles have a component outlet in the back. I found this out the hard way so I have to import all NTSC versions of the Gamecube games I own and use a region-free patch on the Wii in order to use the 480P option.
The notion that Gamecubes, Dreamcasts, OG Xbox, PS2 etc all can't look great on HDTV is fallacy:
Dreamcast supports VGA support. Now most TV's don't have a VGA port so you need a VGA-to-HDMI convertor which in turn upscales the picture to 1080P so it looks even better than straight VGA alone(although it is not TRUE 1080P, it looks alot better than on a CRT tv). Gamecube/og xbox/PS2 are similar in that they support component cables although GC component cables are veryyyyyyyy expensive so the Wii component cables and console are a cheaper choice to enjoy GC games in higher quality.