>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.
It draws one white rectangle per frame over each duck when you pull the trigger and checks if the Zapper can see it
Almost, but not quite. First it blanks the entire screen to solid black and uses that to calibrate the black level of the gun, then it draws a white rectangle over one duck on one frame, then a white rectangle over the other duck on the next frame.
The NES could use this information to determine where the gun was pointing by firing an interrupt at the exact moment when the zapper’s photodiode reached a threshold brightness level above black, and then only register a hit if that occurred while the game was drawing the white rectangle. I think in reality the game didn’t care that much about the timing, only that a rising edge occurred after the fully black frame but before the return to a normal colour frame.
Either way, an LCD doesn’t work because it can’t transition full black to full white within a one frame window. It sometimes works in the 2 duck mode, but it usually records a hit on the wrong duck. In any case, it requires black to white latency less than 16ms