Sprites have nothing to do with it. The same thing would happen with Final Fantasy VII or VHS.
The problem with LCDs (old crap ones anyway) is that the low spec analog signals like composite, s-video, etc have to be digitized and de-interaced and scaled to be usable to the digital portion of the set. This takes a fraction of a second. With a movie you don't notice it since they compensate for the lag by delaying the audio as well. With a game though, its pretty bad. As you said, not a big deal with an RPG, but with a fighting game or something...forget about it.
I'm actually not much of an expert on this since I only have old shit. I'm taking my MVS home for Xmas though, just so I can test it on my friends big new-ish Samsung. I want to see how things have progressed.