I don't believe refresh rate has anything to do with it, or even panel speed. I think all panels are faster than their processor. What's 85hz or 120hz do the output from your console is still 60hz? The Sega Game Gear, like 18 years ago, had an LCD screen that was instant, as did the NEC TurboXpress system years before that. LCD panels are fast enough for anything. Their colour changing speed is a different story, but I don't really care what colour Mario is, I just want him to jump on time!
This is all 100% video processor dependent. LCD monitors got slower before they got faster a few years ago. I've used some late 90s LCDs and they were instant, my 2007 Dell - garbage. Panels have only gotten better since then, processors have only gotten more to do, hence slower.