Depends on how the system works.
If it's like the typical generic ebike/scooter controller, the display has an electronic switch in it that passes the battery voltage it gets back to the controller to power the controller's LVPS and turn it on, after you turn the display's power button on.
It also then commmunicates via serial between controller and display to let you change assist levels from whatever power-on level is. Often that power on level is zero, or no assist, to prevent stuck-throttle or pedalling-take-off accidents, so without the display, the system wont' provide any assist as the level can't be changed from zero.
If yours is instead like some of the scooter/motorcycle systems, the display does nothing except for displaying data, and has no assist levels or other controls routed thru it, so you'd need to find out why the system doesn't turn on (probably other wiring damage somewhere), and then the system would work with or without the display.
If yours is like some of the scooters, the throttle is part of the display, and the throttle is read by the display and coded as serial data to send to the controller, so the system can't do anything without all of these parts working.
Some systems are very proprietary and use serial data between all parts including battery to ensure it's all working and monitor status, etc., and nothing works if everything doesn't work.
Which controller, display, throttle, battery, etc does your system use?