I think a simpler proof is this: If you assume it’s possible for there to be a repeating configuration; which could just be every ball bouncing pack and forth on separate, parallel paths between opposite sides, then it follows there are multiple similar repeating configurations, even if they just involve placement of the balls. There are likely many more complicated repeating configurations, but the simple examples are enough.
Any repeating motion taken by the balls will preclude the taking of any other repeating position not a subset of the first, so all possible paths can’t be taken.
TL/DR: If multiple separate repeating paths exist, only one can randomly show up, since when it does the motion is constrained to repeat forever.