>Lightgun games, for example, rely on very precise timing of the control input vs the CRT raster and simply do not work without a CRT.

Perhaps the most famous light gun game of all time (Duck Hunt on the NES), does not rely on especially precise timing. It draws one white rectangle per frame over each duck when you pull the trigger and checks if the Zapper can see it. LCD latency will probably still break this, but it's not like the later Super Scope for the SNES that actually does track the precise raster position. I expect it would be possible to patch the timing in software to make it work for a specific model of LCD. But even if you did this, the Zapper also includes a bandpass filter at the CRT horizontal retrace rate (about 15kHz) to better reject other light sources, so you'd need to mod it to bypass that, or mod the LCD to strobe the backlight at the right frequency.