The LCD and controller work independently, except that the LCD contains the switch that powers the controller, so when the LCD is off, the controller is off.
The LCD has its own power supply, which is on a connector from the controller, but the controller has no influence on it. It's simply a branch taken off the main battery wire inside the controller, so when the battery supplies power to the main connector on the controller, the power wire for the LCD is always live.
All this means that if the LCD goes off, the most likeky cause is that the power to it is being interrupted either because the battery is faulty or because there's a fault in the LCD.
The first thing to check is that the battery isn't cutting out. Next, check that there's not a bad solder joint where the power wires connect to the LCD's pcb. Wiggle the wires where they go in to see if you can make it go off. The last common cause I can think of is that the cable to the controller is damaged, particularly by zip ties that are so tight that they broke a conductor inside the cable. Loosen off any zip-ties and wiggle the wires in that location to see if you can make it go off.
Occasionally, water gets into LCDs. It can cause all sorts of problems, and it normally ends up killing the LCD. Obviously, that can only happen if you ride in the rain.