With most systems, the display doesn't normally have any control over the system, meaning none of hte control inputs pass thru it to get to the controller. If yours does have the control inputs (PAS, brake, etc) connect to the display instead of the controller, then the display is where the input fault could occur, causing it to tell the controller to do things the user hasn't commanded.
Otherwise, if all your control inputs go to the controller and not the display, then the display shouldn't be able to tell the motor to start.
The exception is systems that have a Walk Mode, where you push / hold a button on the display to cause the motor to run very slowly and steadily to help you walk the bike somewhere, compensating for it's weight, basically. But that wouldn't be a jerky motion, even if activated by a stuck button, unless yours works by just pressing the button and not keeping it held all the time, *and* if holding it turns it on, then keeping it held turns it off, *and* continuing to hold it turns it back on--that could cause it to turn on, off, on, off, in a steady rhythmic pattern determined by how long a button press is required to turn on the mode, then off, then back on.
So the most likely fault is within either the PAS sensor itself, or it's connection to the controller, or within the controller electronics themselves. Disconnecting the PAS sensor will eliminate that and it's cable up to that connector, if the problem continues without it..
Regarding adding a throttle: I don't know what inputs it has available; you'd have to check to see if there are models that have a throttle. If there are, then it means it has the input for it. But you'd have to find out if that's just connecting one up and go, or if you'd have to also reprogram yours with the same firmware that's on the throttle version (which is probably not available, meaning you'd have to change the controller board, and possilby the display if it has matched fw with the controller).
If there are no versions with a throttle, the system is unlikely to have even the capability, so you'd have to change out the controller (and display) for a different controller (like VESC, Phaserunner, etc) that has all the functions you need. Most likely it wont' fit where the present controller is, so you'd have to mount it outside the existing casing and ensure it gets whatever waterproofing is needed for your environment.
Regarding water ingress vs whatever problems it can cause: It's often *after* it has dried out that problems occur, because at that point the corrosion has occured, and it can bridge the tiny closely-spaced connections (especially on MCU pins, under the surface mount chips and parts around it, etc).
Where high voltage is present near low voltage components (like in displays) it's not uncommon for damage to happen when it's still wet, if it's powered-on that way, but this is usually catastrophic.
