I've played modern games on CRTs through HDMI->VGA convertors, and still felt lower latency than LCD. OLEDs will eventually catch up, I think, but LCDs are always going to have some lag.
And it's not the refresh rate, it's the time from input -> display picture updates. With a CRT that can happen during the current field being displayed, but it will take at least one frame for any LCD.
You can update an LCD in the middle of the screen just the way you can for CRT.
The only hard limit to the latency of an LCD panel is the finite amount of time that’s needed to flip the fluid in the LCD cells. There is nothing that requires a full frame delay.
Most LCD monitors have a frame buffer to look back in time for things line overdrive compensation, but you can easily do without. In fact, some of the monitor scaler prototypes that I have worked on were initially direct drive because we hadn’t gotten the DRAM interface up and running yet.
Desktop LCD panels themselves typically don’t have memory, laptop panels do but that’s for power saving reasons (to avoid the power of transferring the data over a high speed link and to allow the source to power down.)
And it's not the refresh rate, it's the time from input -> display picture updates. With a CRT that can happen during the current field being displayed, but it will take at least one frame for any LCD.