The Wii outputs the same signal no matter what monitor it hooks up to. It doesn't "sense" anything about the TV through a video cable, LMAO (technologically impossible).
It sounds like you are plugging a set of composite cables into the wrong ports of a TV with component inputs if it only works on "old" TV's. Component delivers the video over three cables instead of compositing them into one.
Whatever the problem may be, unless your Wii specifically is defective, the issue is on your end. The Wii outputs composite video (or component, if you have the proper cables) just like any other device, it doesn't discriminate.