A lot of screens introduce a frame or two of lag (or even more) that didn't exist back in the day of CRTs. 1 frame is probably okay but anything more starts to be quite noticeable.
Also, the original hardware output probably wouldn't work properly with all games on a modern screen EVEN if the connector exists on your TV. Some of the hardware tricks pulled off by a few games will be garbled by the de-interlacer. Other times, your TV's input simply won't acknowledge the existance of something like 240p.
At this point, short of hauling out an old CRT, it might be more advisable to use a highly accurate emulator like BSNES instead.